# Dr. Ulrich Menne

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 Currently: Professor and Yushan Young ScholarNational Taiwan Normal UniversityTaipei, Taiwan Phone: +886 (0)2 7734 6646(at the National Taiwan Normal University) Office: Room M316(at the National Taiwan Normal University) Postal address: Professor Ulrich MenneDepartment of MathematicsNational Taiwan Normal UniversityNo.88, Sec.4, Tingzhou Rd.Wenshan Dist., TAIPEI CITY 11677TAIWAN(R. O. C.).

News: Courses autumn term 2019:

Research areas: geometric measure theory · differential and convex geometry · linear and nonlinear elliptic partial differential equations · functional analysis.

Further information:

My motto: Theorems are corollaries of understanding.

# Research areas

Geometric measure theory · differential and convex geometry · linear and nonlinear elliptic partial differential equations · functional analysis.

# Professional career

 2018 – Professor and Yushan Young Scholar, Department of Mathematics, National Taiwan Normal University (NTNU), Taipei, Taiwan. 2017 – 2018 Interim Professorship Applied Mathematics for László Székelyhidi (50% FTE), University of Leipzig (UL), Germany; Postdoctoral researcher (50% FTE), Max Planck Institute for Mathematics in the Sciences, Leipzig. 2012 – 2017 Max Planck research group leader of the group Geometric Measure Theory, Max Planck Institute for Gravitational Physics – Albert Einstein Institute (AEI), Potsdam, Germany; Associate Professor for Geometric Analysis, Institute for Mathematics, University of Potsdam (UP); joint appointment with the Max Planck Society and UP. 2009 – 2012 Postdoctoral researcher, AEI. 2008 – 2009 Postdoctoral researcher, Department of Mathematics, ETH Zurich. 2008 Postdoctoral researcher, AEI.

# Education

 2008 PhD in Mathematics, Department of Mathematics, University of Tübingen. $\mathcal C^2$ rectifiability and $Q$ valued functions (supervisors: Reiner Schätzle and Tom Ilmanen). 2005 Diplom-Mathematiker (equivalent to MSc in mathematics), Mathematical Institute, University of Bonn.

# Funding and fellowships

 2018 – Yushan Young Scholarship (Programme of the Ministry of Education in Taiwan to "assist the universities in Taiwan to attract the world's elite"). 2012 – 2017 Funding for the research group Geometric Measure Theory (duration: five years; source: Max Planck Society). 2013 – 2017 Funding for PhD student Mario Santilli (duration: four years; source: Max Planck graduate schools). 2000 – 2005 Fellowship for Diploma studies (duration: 4.5 years; source: State of Bavaria).

# Supervision of graduate students and postdoctoral researchers

## Former postdoctoral researchers at the AEI

• Sławomir Kolasiński (2012 – 2016; now: Adiunkt, University of Warsaw, Poland)
• Anna Sakovich (2014 – 2015; now: Associate Senior Lecturer, Uppsala University, Sweden)
• Ananda Lahiri (2014 – 2017)
• Yangqin Fang (2015 – 2017; now: Associate Professor, Huazhong University, China)

## Former BSc and MSc student at UP

• Christian Scharrer (degrees: 07/2013 and 09/2016; now: PhD student, University of Warwick, UK)

# Key cooperations partners

 2013 – Collaboration with my former team members, Sławomir Kolasiński, Christian Scharrer, and Mario Santilli, AEI (see publications).

# Research visits

 2019 Poland, twelve days: University of Warsaw; host Sławomir Kolasiński. 2017 Switzerland, six months: University of Zurich; host Camillo De Lellis. 2015 UK, three weeks: Universities of Cambridge and Warwick; hosts Costante Bellettini and David Preiss. 2011 US, one month: Universities of Rice, Columbia and Stanford and the Massachusetts Institute of Technology (MIT); hosts: Leobardo Rosales, Lan-Hsuan Huang, Richard Schoen, and Michael Eichmair. 2009 Japan, three weeks: Hokkaido University; Yoshihiro Tonegawa. 2006 Switzerland, seven months: ETH Zurich; Tom Ilmanen.

# Invited presentations at conferences (top five)

 2018 Workshop on Geometric Measure Theory and Minimal Submanifolds, National Center for Theoretical Studies, Taiwan. 2017 Geometric Measure Theory, University of Warwick, UK. 2015 Geometric Measure Theory and Calculus of Variations: theory and applications, Institut Fourier, Grenoble, France. 2013 ERC Workshop on Geometric Measure Theory, Analysis in Metric Spaces and Real Analysis, SNS Pisa, Italy. 2012 Calculus of variations, Mathematical Research Institute Oberwolfach (MFO), Germany.

# Schools and minicourses on my research

 2019 Seven lectures (joint with Mario Santilli), Relating curvatures of geometric measure and convex geometry, Summer School Geometric Measure Theory and Related Fields, Beijing, China. 2018 Six lectures, Sobolev functions and second order elliptic PDEs on varifolds, Seminar on Geometric Measure Theory, Varifolds, and Their Applications, Portland, US. 2016 Five-lecture course, Weakly differentiable functions and Sobolev functions on varifolds, thematic week Geometric Measure Theory, Toulouse, France. 2015 Two double-lectures, Weakly differentiable functions on varifolds, University of Cambridge, UK.

# Invited presentations at seminars (top five)

 2015 University of Warwick, UK. 2013 Charles University, Prague, Czech Republic. 2011 University of Cambridge, UK. 2011 US round trip with talks at MIT and the Universities of Rice, Columbia, and Stanford. 2009 Hokkaido University, Japan.

# Organisation of research seminars and conferences

 2012 – 2017 Co-Organiser (with Theodora Bourni, Klaus Ecker, Mathew Langford, and Jan Metzger), research seminar Topics in geometric analysis; AEI, UP, and the Free University of Berlin. 2012 Co-Organiser (with G. Huisken and N. Wickramasekera), conference Geometric measure theory, AEI. Speakers (seven of 16): Robert Hardt (Rice University), Jan Malý (Charles University Prague), Fernando Marques (IMPA, Rio de Janeiro), Tristan Rivière (ETH Zurich), Leon Simon (Stanford University), Yoshihiro Tonegawa (Hokkaido University), and Brian White (Stanford University).

# Teaching activities for graduate students

 2014 Minicourse, main lecturer and organiser, Plateau type problems using sets, AEI. 2013 – 2016 Reading seminar, organiser, Geometric measure theory, AEI. 2012 Minicourse, lecturer, Almgren’s optimal isoperimetric inequalities, UP. 2011 Minicourse, lecturer, Some aspects of Allard’s regularity theorem, MIT. 2010 Minicourse, lecturer, Federer’s curvature measures, Free University of Berlin.

# Peer review activities and memberships (top five)

 2019 Scientific Member of the National Center for Theoretical Sciences, Taipei. 2015 – 2017 Member of graduate school: Berlin Mathematical School. 2012 – 2015 Member of graduate school: IMPRS for Geometric Analysis, Gravitation and String Theory, Potsdam. 2011 – Reviewer journals (top five): Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Calc. Var. Partial Differential Equations, Geom. Funct. Anal., J. Amer. Math. Soc., and J. Differential Geom. 2009 Reviewer proposal: National Science Foundation (USA).

# Courses for BSc- and MSc-students

 2019 – 2020 Geometric Measure Theory I (lecture), NTNUTopics: Riesz representation theorem, covering theorems, derivatives of measures, locally convex spaces, weak topology, exterior algebra of a vector space, alternating forms, second fundamental form, Grassmann manifold. 2019 – 2020 Topics in Geometric Analysis I (seminar, joint with Chun-Chi Lin), NTNUTopics: Computations and geometric measure theory, functions of bounded variation, topological vector spaces, pointwise differentiability of higher order for sets, rectifiability and approximate differentiability of higher order for sets. 2019 Special topics in analysis (lecture), NTNUTopics: Borel and Suslin sets, symmetric algebra of a vectorspace, polynomial functions, classical, approximate, and pointwise differentiation of higher order, rectifiability of higher order. 2019 Real analysis II (lecture with tutorial), NTNUTopics: Lebesgue spaces, Jensen's inequality, Daniell integrals, linear functionals on Lebesgue spaces, Riesz's representation theorem, Fubini's theorem, Lebesgue measure. 2018 – 2019 Real analysis I (lecture with tutorial, joint with Chun-Chi Lin), NTNUTopics own part: Measures and measurable sets, Borel sets, measurable functions (approximation theorems, spaces of measurable functions), Lebesgue integration (basic properties, limit theorems). 2018 Geometric variational problems (lecture), ULTopics: Multilinear algebra, basic geometric measure theory (Hausdorff densities, Hausdorff distance, Kirszbraun's theorem, relative differentials for closed sets, area formula, rectifiable sets), varifolds. 2017 – 2018 Sets of finite perimeter (seminar, in German), UL.Topics: Whitney's extension theorem, Sobolev functions, functions of bounded variation (isoperimetric inequalities, reduced boundary, criterion for finite perimeter). 2017 – 2018 Introduction to geometric analysis (lecture, in German), UL.Topics: submanifolds of Euclidean space (second fundamental form, covariant differentiation, equations of Gauss and Codazzi), Brunn-Minkowski inequality, monotonicity formula, isoperimetric inequalities, subharmonic functions. 2016 – 2017 Mathematics for business information systems (lecture with tutorial, in German), UP.Topics: basics of logic, set theory, number systems, linear algebra, and analysis. 2016 Surfaces in analysis, geometry and physics (lecture series with tutorial, joint with Christian Bär, Jan Metzger, and Sylvie Paycha), UP.Topics own part: BV-functions (convolution, Sobolev- and Poincaré-inequalities), geometric variational problems. 2015 – 2016 Topics in elliptic partial differential equations (lecture with tutorial), UP.Topics: Whitney's extension theorem, rectifiability of higher order for functions, Sobolev functions, ellipticity, pointwise differentiability of solutions for linear elliptic partial differential equations. 2015 Real analysis (lecture with tutorial, in German), UP. Topics: covering theorems of Besicovitch and Vitali; differentiation theory of Radon measures: existence, Lebesgue points, densities, approximate continuity; curves of finite length; theorems of Rademacher and Stepanoff. 2014 – 2015 Borel sets and Suslin sets (seminar, in German), UP.Topics: Borel sets, spaces of sequences, Suslin sets; measures, measurable sets, regular measures, nonmeasurable sets, Borel regular measures; measurability of Suslin sets. 2013 – 2014 Partial Differential Equations (lecture with tutorial), UP.Topics: properties of harmonic functions; multlinear algebra; differentials of higher order; convolution; Sobolev- and Hölder-spaces; $L^2$ theory: Lax Milgram, Gårding-Ungleichung, existence, strong solutions. 2013 Introduction to geometric measure theory (lecture with tutorial, in German), UP.Topics: Hausdorff measures, Cantor sets, Steiner symmetrisation, Kirszbraun's theorem; submanifolds of Euclidean spaces; rectifiable sets; area formula on rectifiable sets. 2012 – 2013 Real analysis (lecture with tutorial, in German), UP.Topics: see the course real analysis above. 2012 Elements in measure theory (seminar, in German), UP.Topics: basics of outer measures, Carathéodory's criterion, theorems of Lusin and Egoroff, Daniell integrals, Riesz-Radon representation theorem, duality of Lebesgue spaces.

# Summary for researchers in geometric analysis

Abstract: My research pertains to the fields of calculus of variations and to elliptic partial differential equations (elliptic PDEs). The common theme of my contributions is geometric measure theory – the natural language for many geometric variational problems. For instance, for generalised submanifolds with mean curvature (and integer multiplicity), I have proven the existence of a geodesic distance and of a second fundamental form, and I have laid the foundation to study PDEs thereon (see [4], [6], and [8]). My research programme on regularity in geometric measure theory is carefully tailored to systematically approach a key open problem and to provide new methods for elliptic PDEs and the calculus of variations.

## Introduction

A central topic of my research are general minimal surfaces of arbitrary dimension and codimension in Riemannian manifolds. There is an extremely rich theory of two-dimensional minimal surfaces. For surfaces of dimension greater than two, the appropriate class to prove existence of general minimal surfaces is formed by integral varifolds (see Almgren [Alm65]). They consist of a set, that admits measure-theoretic tangent planes almost everywhere, together with a locally summable function with values in the positive integers, called multiplicity or density; area in this context refers to area of the underlying set weighted with the density; see [12] for a brief exposition of mine on varifolds. Classically, regularity theory aims at representing the varifold by a continuously differentiable submanifold outside a set whose size can be estimated. In this regard, Allard has shown the existence of an open dense regular set for any stationary (i.e., mean curvature zero) integral varifold (see [All72]). This does not imply that the singular set has measure zero. In fact, it is a key open problem whether all stationary integral varifolds are regular almost everywhere. Much more is known if the varifold is stable (see Wickramasekera [Wic14]) or corresponds to an area-minimising current (see, e.g., Federer [Fed70], Hardt and Simon [HS79], and Almgren [Alm00]).

## My long term goals

When I started research in geometric measure theory as a PhD student, Allard’s afore-mentioned result had already found a very large number of applications and its method of proof (originating from De Giorgi and Almgren) had provided the basis for partial regularity theory in elliptic PDEs and the calculus of variations. Since then, the problem of possible almost everywhere regularity of stationary integral varifolds has served as guide for my research.

Over the years, I have developed a research programme approaching this question in three stages, each stretching over about five years at least. It is based on the investigation of the larger class of integral varifolds satisfying p-th power summability conditions on their mean curvature (1 ≤ p ≤ ∞). Allard’s result is phrased in this class, but almost everywhere regularity dramatically fails therein (see Brakke [Bra78, 6.1]). In the first stage, I have systematically investigated almost everywhere regularity properties in the above class (see [1], [2], [3], [4], and [7]). In the second stage which is nearing completion, I have already significantly enlarged the set of tools from geometric analysis available for the study of an even larger class of varifolds (see [5], [6], [8], [10], [11], and [13]). In the third stage, almost everywhere regularity properties of stationary integral varifolds, that are not shared by all twice continuously differentiable submanifolds, shall be investigated (see [9] for a first proof of concept).

My research addresses one of the pivotal challenges in geometric measure theory: higher multiplicity. Progress therein will be absorbed into topics in differential geometry (e.g., minimal surfaces), in geometric analysis (e.g., mean curvature flow, and Willmore surfaces), and in mathematical models in the natural sciences (e.g., soap bubbles, cell membranes, and phase transitions). It also involves developing new tools for elliptic PDEs (see below).

## Regularity of generalised submanifolds with mean curvature (completed)

The equation satisfied by integral varifolds with p-th power summable mean curvature may be seen as non-graphical, higher multiplicity, quasi-linear analogue of the Poisson equation. Therefore, the goal in this first stage was to determine which parts of the classical Calderón Zygmund theory carry over.

### Geometric Sobolev Poincaré inequalities

Initially, even the degree to which such integral varifolds share the properties of weakly differentiable (multiple-valued) functions was insufficiently understood. In my PhD thesis (see [1] and [2]), this question was comprehensively answered; including optimal geometric Sobolev Poincaré inequalities involving the tilt-excess of type Lq, that is, the slope of the tangent plane with respect to a reference plane, measured in a Lebesgue space with exponent q.

### Pointwise decay estimates of Campanato type

Allard’s regularity theorem is based on uniform decay estimates of the tilt-excess of L2 type. In the presence of higher multiplicity, only pointwise (rather than uniform) decay estimates may be obtained. To clean the picture, the partial results implicit in Brakke [Bra78, § 5] were improved to optimal results in all but one case in [3]; the last case was treated jointly with Kolasiński in [7].

### Existence of a second fundamental form

The decay estimates of [3] are necessarily limited to orders strictly below two. In [4], I have established the existence of a second fundamental form whose trace equals the variationally defined mean curvature almost everywhere for any integral varifold of locally bounded first variation (i.e., for p = 1). In [Sch04], Schätzle had shown the same conclusion to hold in codimension one, provided that p exceeds the dimension of the varifold. For the generalisation to higher codimension and the substantial weakening regarding the summability of the mean curvature, I had to develop completely new tools yielding novel results even for the ordinary Poisson equation.

### No Gehring improvement and no second order Lebesgue estimates

The results of [4] entail, almost everywhere, second order decay rates (but no estimates) for the tilt-excess of L2 type. In [7], we have shown that, for p = ∞ (unlike for area-minimising rectifiable currents (see Almgren [Alm00, § 3])), there is no Gehring improvement to L2+ε for any ε > 0, and that the second fundamental form does not belong to L1+ε for any ε > 0; both not even near almost every point. Since zero order quantities were long understood, these results complete the first stage of my programme.

## Geometric analysis on generalised submanifolds with mean curvature (in progress)

The initial motivation for the second stage was the question whether, for any compact, nonempty integral varifold with mean curvature, the integral of the mean curvature with dimensionally critical power is smallest for spheres. In codimension one (and varifold dimension at least two), I have solved this question in the affirmative by using the second fundamental form of [4] and establishing a Harnack type inequality for Lipschitzian solutions of the Poisson equation involving the Laplace Beltrami operator on the varifold (see the extended abstract [5]). This indicates the potential of a systematic study of second order, divergence form elliptic PDEs on varifolds and their geometric applications.

### Weakly differentiable functions and Sobolev functions on varifolds (completed)

For smooth submanifolds, Sobolev functions of any order are readily defined using charts. For integral varifolds of locally bounded first variation, in contrast, the obvious strategies (i.e., employing the usual integration-by-parts identities involving mean curvature, taking completions in Lebesgue spaces based on weighted area, or treating the varifold as metric measure space) all fail for different reasons. In fact, I have shown in [6] that, even on stationary integral varifolds, there is no satisfactory class, that is closed with respect to addition and post-composition, and that is well-behaved with respect to decompositions of the underlying varifold. Instead, I have built a coherent theory of weakly differentiable functions (a purely geometric, non-additive class of functions) and of Sobolev functions (a linear subclass for functional analytical considerations), in particular, on such varifolds (see [6], [8], and [10], totalling 176 pages). The theory equally applies to both classes, and comprises a variety of Sobolev Poincaré type embeddings, Rellich type theorems, compact embeddings into spaces of continuous functions, and pointwise differentiability results both of approximate and integral type, as well as coarea formulae.

### Geodesic distance associated with varifolds (completed)

Geodesics are a fundamental tool in Riemannian geometry not previously available for (even stationary) integral varifolds. In [6] and [8], I have proven that the geodesic distance associated with an integral varifold with critical summability of the mean curvature (i.e., p equal to the dimension m of the varifold) is a continuous, real-valued Sobolev function on each open connected component of its support. The subtlety of this results is illustrated by the fact that the geodesic distance to a fixed point is a possibly non-Hölder continuous Sobolev function, whose weak derivative has modulus one almost everywhere. Despite p = m is optimal in general, for compact, indecomposable, integral varifolds, existence of the geodesic distance holds for p = m − 1; in fact, jointly with Scharrer (extending his MSc thesis [Sch16]), we established this fact as consequence of a novel type of Sobolev-Poincaré inequality in [13].

### Second order, divergence form elliptic PDEs on varifolds (in progress)

Having a coherent theory of Sobolev functions – including the central geometric inequalities of Sobolev and Poincaré type – at one’s disposal, the next obvious task is to build basic PDE theory on varifolds. This includes bringing the Harnack type inequality (and the resulting strong maximum principle) for Lipschitzian solutions of the Poisson equation on varifolds (which led to [5]) to their natural generality.

### The normal bundle of varifolds and a sharp geometric inequality (in progress)

For smooth surfaces, the Gauss map in codimension one and the unit normal bundle in general dimensions are important tools to study these surfaces by means of their image in the sphere. In [11], I obtain jointly with Santilli (as PhD student in my group), for arbitrary closed sets, a natural stratification of the part, where the set can be touched from at least one direction (generalising the case of convex sets, see Alberti [Alb94]), and the existence of a second fundamental form on each stratum. In [San17], this led to a powerful criterion (in terms of a Lusin property and the mean curvature), when the whole set (not only its stratifiable part) is almost equal to a single stratum.

### Non-integral case

Apart of [5], all varifold results of this section (i.e., [6], [8], [10], and [Sch16]) in fact apply to possibly non-integral varifolds satisfying a uniform lower density bound (or less).

## Regularity of generalised minimal surfaces (starting phase)

After developing regularity theory up to second order (which is the natural limit under summability conditions on the mean curvature) in the first stage, and with the large toolbox developed in the second stage at hand, the third stage shall consider higher order properties of stationary integral varifolds.

### Higher order differentiability of sets (completed)

To establish a solid base, I have developed, for orders 1 ≤ γ < ∞, a theory of pointwise differentiability of order γ for arbitrary subsets of Euclidean space (see [9]). The concept is characterised by a limit procedure involving inhomogeneously dilated sets. Despite local graphical representability is not implied, the same theorems as for functions could be derived, including rectifiability of order γ, and a Rademacher-Stepanoff theorem. The usefulness of this concept is indicated by the fact that the support of stationary integral varifolds, where it intersects a given plane, is almost everywhere pointwise differentiable of every finite order (see [9]).

### Higher order pointwise differentiability of stationary integral varifolds (planned)

To remove the proviso (intersection by a given plane), pointwise decay estimates of higher non-integer order shall be developed.

### Higher order pointwise differentiability theory for distributions and non-linear elliptic PDE (started)

Locally uniform decay estimates of order below two have been transferred with very large success from geometric measure theory to non-linear elliptic PDE. Therefore, investigating a similar transfer for the pointwise decay estimates of higher non-integer order (see above) would be a natural project; as preparation for both this and the varifold case, I established an analogous theory for distributions in [14].

## References by other authors

 [Alb94] Giovanni Alberti. On the structure of singular sets of convex functions. Calc. Var. Partial Differential Equations, 2(1):17–27, 1994. [All72] William K. Allard. On the first variation of a varifold. Ann. of Math. (2), 95:417–491, 1972. [Alm65] F. J. Almgren, Jr. The theory of varifolds. Mimeographed Notes. Princeton University Press, 1965. [Alm00] Frederick J. Almgren, Jr. Almgren’s big regularity paper, volume 1 of World Scientific Monograph Series in Mathematics. World Scientific Publishing Co. Inc., River Edge, NJ, 2000. Q-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2, With a preface by Jean E. Taylor and Vladimir Scheffer. [Bra78] Kenneth A. Brakke. The motion of a surface by its mean curvature, volume 20 of Mathematical Notes. Princeton University Press, Princeton, N.J., 1978. [Fed70] Herbert Federer. The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension. Bull. Amer. Math. Soc., 76:767–771, 1970. [HS79] Robert Hardt and Leon Simon. Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. of Math. (2), 110(3):439–486, 1979. [San17] Mario Santilli. Curvature of closed subsets of Euclidean space and minimal submanifolds of arbitrary codimension, 2017. arXiv:1708.01549v1. [Sch04] Reiner Schätzle. Quadratic tilt-excess decay and strong maximum principle for varifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 3(1):171–231, 2004. [Sch16] Christian Scharrer. Relating diameter and mean curvature for varifolds. MSc thesis, University of Potsdam, 2016. [Wic14] Neshan Wickramasekera. A general regularity theory for stable codimension 1 integral varifolds. Ann. of Math. (2), 179(3):843–1007, 2014.

# Introductory material on varifolds (for general mathematicians)

A varifold is a very general, measure-theoretic model of a possibly highly singular generalised submanifold with or without boundary in an Euclidean space or in a Riemannian manifold. Varifold theory is particularly well suited for problems leading to surfaces, that admit a variationally defined mean curvature vector. In contrast to other models of singular surfaces, the area-functional on varifolds is continuous rather than just lower semicontinuous.

A particular important class are integral varifolds. This subclass is the strong closure of the set of varifolds which are finite sums of varifolds corresponding to measurable subsets of continuously differentiable submanifolds. It enjoys favourable compactness properties under integral mean curvature bounds. In particular, Almgren proved in 1965 that, within any nonempty compact, smooth Riemannian manifold, there exists (for each intermediate dimension) a nonzero integral varifold that is stationary with respect to the area-integrand (i.e., there exists a nonzero generalised minimal surface).

In general, the structure of stationary integral varifolds is still quite baffling. Nevertheless, I have made substantial progress in understanding their extrinsic geometry (i.e., their regularity properties) as well as their intrinsic geometry (i.e., their geodesic distance and properties of spaces of weakly differentiable functions on them). This in turn rests on the study of more basic geometric properties of varifolds and it is naturally linked to the study of non-smooth subsets of Euclidean space.

The basic definitions and theorems on such varifolds, are reviewed in a short exposition of mine by means of 20 examples, assuming mainly familiarity with basic measure theory.

# Summary for general audience

Generally speaking, my research aims at the understanding of the complex local structure of surfaces occurring in many models from the natural sciences. In this regard, a mathematical surface may correspond to a variety of different physical objects: for instance, soap films, horizons of black holes, membranes of cells, and boundaries between different phases of a material, or between different grey levels in a digitally reconstructed image.

It is the power of mathematical abstraction, that allows to devise a model of surfaces capable of covering all these cases at once and to derive theoretical conclusions (e.g., regularity results), that typically are applicable in all of these settings. By a regularity result, one means a mathematical theorem, that says, that a surface satisfying a given optimality condition admits a simpler (i.e., more regular) local description, than an arbitrary surface in its class.

The study of models of surfaces with a complex local structure pertains to the field of geometric measure theory. The success of this theory does not only stem from the many applications, that its results have found in other larger fields within mathematics (e.g., differential geometry, geometric analysis, and mathematical models in the natural sciences), but also from the new methods that it has contributed to the big fields of partial differential equations and the calculus of variations.

The core motivation for my research is to make progress towards a fundamental regularity question formulated by Allard in 1972. This question concerns the local structure of surfaces in a particularly versatile class of surfaces (namely, integral varifolds) under the natural optimality condition for this class. This class of surfaces is employed in models of all of the above-mentioned physical objects.

My research programme in this direction consists of three, partially overlapping stages. In the first two stages, I systematically investigate which properties of their regular cousins (namely, twice continuously differentiable submanifolds) are shared by integral varifolds in case they satisfy the optimality condition. More precisely, in the first stage (which is now completed), I studied regularity theorems for these integral varifolds, and in the second stage (which is nearing completion), I transfer mathematical machinery previously only available for the regular cousins to the relevant classes of varifolds. In contrast, the third stage, which is currently in the starting phase, aims at establishing regularity properties that exceed those of their regular cousins.

## Publikationen

Differentiability theory for distributions and subsets of Euclidean space

U. Menne
[14] Pointwise differentiability of higher order for distributions, 30 pages.
Submitted, 2018.
ArXiv: arXiv:1803.10855v1 [math.FA].
Abstract: For distributions, we build a theory of higher order pointwise differentiability comprising, for order zero, Łojasiewicz's notion of point value. Results include Borel regularity of differentials, higher order rectifiability of the associated jets, a Rademacher-Stepanov type differentiability theorem, and a Lusin type approximation. A substantial part of this development is new also for zeroth order. Moreover, we establish a Poincaré inequality involving the natural norms of negative order of differentiability. As a corollary, we characterise pointwise differentiability in terms of point values of distributional partial derivatives.

U. Menne, M. Santilli
[11] A geometric second-order-rectifiable stratification for closed subsets of Euclidean space, 14 pages.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19(3):1–14, 2019.
ArXiv: arXiv:1703.09561v2 [math.CA].
Abstract: Defining the m-th stratum of a closed subset of an n dimensional Euclidean space to consist of those points, where it can be touched by a ball from at least n−m linearly independent directions, we establish that the m-th stratum is second-order rectifiable of dimension m and a Borel set. This was known for convex sets, but is new even for sets of positive reach. The result is based on a sufficient condition of parametric type for second-order rectifiability.

U. Menne
[9] Pointwise differentiability of higher order for sets, 31 pages.
Ann. Global Anal. Geom., 55(3), 591–621, 2019.
Springer Nature SharedIt: https://rdcu.be/bgUqa. DOI: 10.1007/s10455-018-9642-0. ArXiv: arXiv:1603.08587v2 [math.DG].
Abstract: The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that differentials are Borel functions, higher order rectifiability of the set of differentiability points, and a Rademacher result. One concept is characterised by a limit procedure involving inhomogeneously dilated sets. The original motivation to formulate the concepts stems from studying the support of stationary integral varifolds. In particular, strong pointwise differentiability of every positive integer order is shown at almost all points of the intersection of the support with a given plane.

Survey on varifolds suitable for beginning graduate students

U. Menne
[12] The concept of varifold, 5 pages.
Notices Amer. Math. Soc., 64(10):1148–1152, 2017.
DOI: 10.1090/noti1589. ArXiv: arXiv:1705.05253v4 [math.DG].
Abstract: We survey – by means of 20 examples – the concept of varifold, as generalised submanifold, with emphasis on regularity of integral varifolds with mean curvature, while keeping prerequisites to a minimum. Integral varifolds are the natural language for studying the variational theory of the area integrand if one considers, for instance, existence or regularity of stationary (or, stable) surfaces of dimension at least three, or the limiting behaviour of sequences of smooth submanifolds under area and mean curvature bounds.

Partial differential equations on singular surfaces with mean curvature – foundations

U. Menne
[8] Sobolev functions on varifolds, 50 pages.
Proc. Lond. Math. Soc. (3), 113(6):725–774, 2016.
Free access link (currently, out of order; fix under way). DOI: 10.1112/plms/pdw023. ArXiv: arXiv:1509.01178v3 [math.CA].
Abstract: This paper introduces first-order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally non-linear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to satisfy a uniform lower density bound and a dimensionally critical summability condition on its mean curvature, the following statements hold. Firstly, continuous and compact embeddings of Sobolev spaces into Lebesgue spaces and spaces of continuous functions are available. Secondly, the geodesic distance associated to the varifold is a continuous, not necessarily Hölder continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, then the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding many new results for those classes as well. Suitable versions of the embedding results obtained for Sobolev functions hold in the larger class of generalised weakly differentiable functions.

U. Menne
[6] Weakly differentiable functions on varifolds, 112 pages.
Indiana Univ. Math. J., 65(3):977–1088, 2016.
DOI: 10.1512/iumj.2016.65.5829. ArXiv: arXiv:1411.3287v1 [math.DG].
Abstract: The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration-by-parts identities for certain compositions with smooth functions. In this class, the idea of zero boundary values is realised using the relative perimeter of superlevel sets. Results include a variety of Sobolev Poincaré-type embeddings, embeddings into spaces of continuous and sometimes Hölder-continuous functions, and pointwise differentiability results both of approximate and integral type as well as coarea formulae. As a prerequisite for this study, decomposition properties of such varifolds and a relative isoperimetric inequality are established. Both involve a concept of distributional boundary of a set introduced for this purpose. As applications, the finiteness of the geodesic distance associated with varifolds with suitable summability of the mean curvature and a characterisation of curvature varifolds are obtained.

Regularity of singular surfaces with mean curvature

S. Kolasiński, U. Menne
[7] Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds, 56 pages.
NoDEA Nonlinear Differential Equations Appl., 24:Art. 17, 56, 2017.
DOI: 10.1007/s00030-017-0436-z. ArXiv: arXiv:1501.07037v2 [math.DG].
Abstract: This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere of the super-quadratic tilt-excess are obtained. These examples are optimal in terms of the dimension of the varifold and the exponent of the integrability condition in most cases, for example if the varifold is not two-dimensional. These examples also demonstrate that within the scale of Lebesgue spaces no local higher integrability of the second fundamental form, of an at least two-dimensional curvature varifold, may be deduced from boundedness of its generalised mean curvature vector. Amongst the tools are Cartesian products of curvature varifolds.

U. Menne
[4] Second order rectifiability of integral varifolds of locally bounded first variation, 55 pages.
J. Geom. Anal., 23(2):709–763, 2013.
DOI: 10.1007/s12220-011-9261-5. ArXiv: arXiv:0808.3665v3 [math.DG].
Abstract: It is shown that every integral varifold in an open subset of Euclidean space whose first variation with respect to area is representable by integration can be covered by a countable collection of submanifolds of the same dimension of class 2 and that their mean curvature agrees almost everywhere with the variationally defined generalized mean curvature of the varifold.

U. Menne
[3] Decay estimates for the quadratic tilt-excess of integral varifolds, 83 pages.
Arch. Ration. Mech. Anal., 204(1):1–83, 2012.
DOI: 10.1007/s00205-011-0468-1. ArXiv: arXiv:0909.3253v3 [math.DG].
Abstract: This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with their first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic tilt-excess are established for those varifolds. The results are optimal in terms of the dimension of the varifold and the exponent of the Lebesgue space in most cases, for example if the varifold is not two-dimensional.

Basic geometric properties of singular surfaces with mean curvature

U. Menne, C. Scharrer
[13] A novel type of Sobolev-Poincaré inequality for submanifolds of Euclidean space, 35 pages.
In revision (to include one or more applications), 2017.
ArXiv: arXiv:1709.05504v1 [math.DG].
Abstract: For functions on generalised connected surfaces (of any dimensions) with boundary and mean curvature, we establish an oscillation estimate in which the mean curvature enters in a novel way. As application we prove an a priori estimate of the geodesic diameter of compact connected smooth immersions in terms of their boundary data and mean curvature. These results are developed in the framework of varifolds. For this purpose, we establish that the notion of indecomposability is the appropriate substitute for connectedness and that it has a strong regularising effect; we thus obtain a new natural class of varifolds to study. Finally, our development leads to a variety of questions that are of substance both in the smooth and the nonsmooth setting.

U. Menne, C. Scharrer
[10] An isoperimetric inequality for diffused surfaces, 16 pages.
Kodai Math. J., 41(1):70–85, 2018.
DOI: 10.2996/kmj/1521424824. ArXiv: arXiv:1612.03823v2 [math.DG].
Abstract: For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of varifold theory in the study of diffused surfaces.

U. Menne
[2] A Sobolev Poincaré type inequality for integral varifolds, 40 pages.
Calc. Var. Partial Differential Equations, 38(3-4):369–408, 2010.
DOI: 10.1007/s00526-009-0291-9. ArXiv: 0808.3660v2 [math.DG].
Abstract: In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.

U. Menne
[1] Some applications of the isoperimetric inequality for integral varifolds, 23 pages.