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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).

Academic beneficiaries

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 [Zaj79]), 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

[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.
[Zaj79] Luděk Zajíček. On the differentiation of convex functions in finite and infinite dimensional spaces., Czechoslovak Math. J., 29(104)(3):340–348, 1979.

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(研究摘要)

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.

一般而言,我的研究目標在於了解局部的複雜曲面結構,它出現在許多自然科學的模型。在這方面,數學曲面可能和許多各式各樣的物理物件有關連:例如,肥皂泡膜、黑洞視界、細胞膜以及物質不同相的界面,或者經數位化重建的影像中不同灰階之間的界面。

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.

正是數學抽象化的力量,讓我們設想出一個曲面模型,適用剛剛提到的所有實例,並從理論上得出結果(例如:regularity結果),足以代表性地應用於全部情境。所謂regularity結果,指的是一則數學定理,它說,當曲面滿足某個最佳條件時,可以給出更簡單(也就是更regular)的局部描述,在同類型的曲面中,這個特性並非任意可見。

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.

具有局部複雜結構的曲面模型之研究與幾何測度論領域有關。幾何測度論的成功,不僅因為它的結果有許多應用,出現在其他更寬廣的數學領域(例如:微分幾何、幾何分析以及自然科學中的數學模型),也因為它的方法新穎,為偏微分方程與變分學這個重大領域作出貢獻。

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.

我的研究的核心動機在於推動基礎的regularity問題的進展,這個問題是Allard於1972年所提出的。這個問題關心某一類用途特別廣的曲面(也就是可積varifolds)在符合自然最佳條件時所具有的局部結構。這類曲面用於前面提到的所有物理物件的模型之中。

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 counterparts (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 counterparts 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 counterparts.

我在這方面的研究計畫包含了三個不同但部分重疊的階段。在最初兩個階段,我系統性地探究可積varifolds在符合最佳條件時,和他們的regular同類(也就是二階連續可微子流形)有哪些共同的特性。更精確地說,在第一階段(目前已完成),我研究了這些可積varifolds的regularity定理。而在第二階段(目前已接近完成),我把之前只適用於regular 同類的數學機制推廣到相關類型的varifolds之上。往另一個方向發展,是目前剛起步的第三階段的目標,試圖建立超越他們的regular同類所具備的regularity性質。