Schedule for: 22w5181 - Integral and Metric Geometry (Online)

The curvature integrals that arise in the tube formula for a Riemannian manifold M embedded in euclidean space are remarkable for at least two reasons: they are known to be artifacts of the curvature tensor of M, hence intrinsic to the metric structure of M (Weyl 1939); and they may be constructed, as signed measures, for a broad class of “singular spaces” beyond smooth submanifolds– for example PL spaces, or sets with positive reach (Federer 1959). An irresistible circle of questions arises, starting with: are these measures intrinsic to the metric structure of these singular spaces? which singular spaces admit them? how can they be constructed? Despite a great deal of recent progress on related questions, the core issues remain stubbornly resistant to analysis. I will review some of the things that have been done, discuss some of the difficulties, and propose some concrete problems that seem to me to be central. Our present point of view is that the idea of valuations will be key to future progress.

I will review the notion of an integral current space which was defined jointly with Stefan Wenger applying Ambrosio-Kirchheim's notion of an integral current on a metric space.   I will present properties of these spaces proven jointly with Jacobus Portegies.   I will pose open questions that are of interest to those hoping to prove Misha Gromov's Compactness Conjecture concerning the limits of non-collapsing sequences of Riemannian manifolds with nonnegative scalar curvature including questions posed by Raquel Perales at a prior meeting.   If there is time, I will also present key examples related to the conjecture constructed with Jorge Basilio, Jozef Dodziuk, Wenchuan Tian and Changliang Wang.   

In this seminar I will discuss the so called "zonoid algebra", a construction introduced in a recent work (joint with Breiding, Bürgisser and Mathis) which allows to put a ring structure on the set of zonoids (i.e. Hausdorff limits of Minkowski sums of segments). This framework gives a new perspective on classical objects in convex geometry, and it allows to introduce new functionals on zonoids, in particular generalizing the notion of mixed volume. Moreover this algebra plays the role of a probabilistic intersection ring for compact homogeneous spaces. Joint work with P. Breiding, P. Bürgisser and L. Mathis

A Finsler manifolds can be considered as a manifold that carries a convex body in each tangent space. In this talk I'll survey some applications that result from applying convex-geometric constructions and inequalities to the study of Finsler manifolds.

Isoperimetric inequalities measure how difficult it is to fill (Lipschitz) cycles in a given space by (Lipschitz) chains of one dimension higher, and they are important in many branches of mathematics. In this talk, I will discuss relationships between isoperimetric inequalities and certain extension properties of the underlying space, especially Lipschitz connectivity and coning inequalities. In many cases, such as Banach spaces and non-positively curved metric spaces, it is easy to establish Lipschitz connectivity or a coning inequality, but harder to obtain an isoperimetric inequality. I will show that Lipschitz connectivity implies Euclidean isoperimetric inequalities and that Euclidean isoperimetric inequalities imply coning inequalities when the underlying space has finite Nagata dimension. These results are consequences of a more general theorem about isoperimetric subspace distortion which asserts that if a metric space X has finite Nagata dimension and is Lipschitz k-connected or admits Euclidean isoperimetric inequalities up to dimension k then k-dimensional cycles in X can be filled almost as efficiently in X as in any space into which X embeds isometrically. Based on joint work with Giuliano Basso and Robert Young.

We will present an analog of Santaló point for the area of the boundary of a convex body in a normed space, and explain why they exist in general, and under which conditions they are unique. We will also provide a characterization of such a point when the normed space is Minkowski. This is joint work with G.Solanes and K.Tzanev.

The Alesker-Bernig-Schuster theorem describes the decomposition of the space of translation-invariant continuous valuations into irreducible representations of the special orthogonal group. We construct an explicit set of the corresponding highest weight vectors and characterize important natural operations on valuations (pullback, pushforward, Fourier transform, Alesker-Poincar\'e pairing, and Lefschetz operator) in terms of their action on these vectors. As an application, we prove a version of the Hodge-Riemann relations for valuations which has previously been observed to imply new geometric inequalities between mixed volumes of convex bodies. Joint work with Thomas Wannerer.

Closed, embedded constant mean curvature (CMC) hypersurfaces in the Euclidean, spherical and hyperbolic spaces have been classified as round spheres in a nowadays classical and famous paper by Alexandrov from the 1960s. In a class of warped product spaces, such as the de-Sitter-Schwarzschild spaces, such a classification is due to Simon Brendle and about 10 years old. The stability question for these results ask, whether closed and embedded "almost" CMC hypersurfaces must be Hausdorff-close to geodesic spheres. While in the Euclidean, spherical and hyperbolic spaces this problem has been studied with very general results by Magnanini-Poggesi, Ciraolo-Vezzoni and several others, in this talk we want to present the proof of the corresponding counterpart in a class of warped product spaces with non-constant sectional curvature. This is joint work with Chao Xia (Xiamen University).

The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$, known to the ancient Greeks in dimensions two and three, states that among all sets (``bubbles") of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the sphere $\mathbb{S}^n$ and on Gauss space $\mathbb{G}^n$. Furthermore, one may consider the ``multi-bubble" partitioning problem, where one partitions the space into $q \geq 2$ (possibly disconnected) bubbles, so that their total common surface-area is minimal. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to $q=2$; the case $q=3$ is called the double-bubble problem, and so on. In 2000, Hutchings, Morgan, Ritor\'e and Ros resolved the Double-Bubble conjecture in Euclidean space $\mathbb{R}^3$ (and this was subsequently resolved in $\mathbb{R}^n$ as well) -- the optimal partition into two bubbles of prescribed finite volumes (and an exterior unbounded third bubble) which minimizes the total surface-area is given by three spherical caps, meeting at 120-degree angles. A more general conjecture of J.~Sullivan from the 1990's asserts that when $q \leq n+2$, the optimal Multi-Bubble partition of $\mathbb{R}^n$ (as well as $\mathbb{S}^n$) is obtained by taking the Voronoi cells of $q$ equidistant points in $\mathbb{S}^{n}$ and applying appropriate stereographic projections to $\mathbb{R}^n$ (and backwards). In 2018, together with Joe Neeman, we resolved the analogous Multi-Bubble conjecture on the optimal partition of Gauss space $\mathbb{G}^n$ into $q \leq n+1$ bubbles -- the unique optimal partition is given by the Voronoi cells of (appropriately translated) $q$ equidistant points. In this talk, we will describe our approach in that work, as well as recent progress on the Multi-Bubble problem on $\mathbb{R}^n$ and $\mathbb{S}^n$. In particular, we show that minimizing partitions are always spherical when $q \leq n+1$, and we resolve the latter conjectures when in addition $q \leq 6$ (e.g. the triple-bubble conjecture in $\mathbb{R}^3$ and $\mathbb{S}^3$, and the quadruple-bubble conjecture in $\mathbb{R}^4$ and $\mathbb{S}^4$). Based on joint work (in progress) with Joe Neeman.

An old problem by Banach asks whether a normed vector space V is necessarily Euclidean if all its n-dimensional linear subspaces are isometric for a fixed n between 1 and dim(V). In geometric terms, the question is whether every centered convex body all whose n-dimensional central cross-sections are affine equivalent, is an ellipsoid. This is proven in some dimensions and remains unknown in others. I will speak about various approaches to the problem and a new advance, joint with D.Mamaev and A.Nordskova, solving the case $n=3$.

The Lipschitz-Killing invariants, discovered by Weyl in his tube formula, are among the most fundamental riemannian invariants. Remarkably, they can be extended to several classes of singular subspaces of a riemannian manifold. Under this form, they belong to a type of functionals called valuations, and they constitute a natural generalization of the intrinsic volumes from convex geometry. I will present a joint project with Andreas Bernig and Dmitry Faifman where we extend the Lipschitz-Killing valuations to the pseudo-riemannian setting. The main difficulty is the appearance of diverging integrals related to the existence of light directions. Our approach, based on distributions, has led in particular to a Gauss-Bonnet formula for metrics of changing signature, and to Crofton formulas in pseudo-riemannian space-forms.

Appearing as coefficients in Weyl’s famous tube formula, Lipschitz–Killing valu- ations (or intrinsic volumes) form an important subalgebra of the space of valuations on a Riemannian manifold. Recently, J. Fu and T. Wannerer showed that this al- gebra is indeed characterized by its invariance under pullback with respect to any isometric immersion between two Riemannian manifolds. In this talk, we consider the dual operation of taking pushforwards of intrinsic volumes with respect to Riemannian submersions. In particular, we explicitly cal- culate the pushforwards of intrinsic volumes on the (2n + 1)-dimensional Euclidean sphere to the n-dimensional complex projective space via the Hopf fibration, which answers a question by J. Fu. This is joint work in progress with T. Wannerer.

We will prove a sharp generalization of Banchoff and Pohl's isoperimetric inequality in complete, simply connected Riemannian manifolds of non-positive sectional curvature. We will also prove a sharp, quantitative version of an isoperimetric inequality of Yau in spaces of negative curvature and a modified version of Croke's sharp 4-dimensional isoperimetric inequality. We will discuss the relationship between these results and the Cartan-Hadamard conjecture, which states that complete, simply connected Riemannian manifolds of non-positive curvature satisfy the Euclidean isoperimetric inequality.

Any Riemannian manifold has a canonical collection of valuations (finitely additive measures) attached to it, known as the intrinsic volumes or Lipschitz-Killing valuations. They date back to the remarkable discovery of Weyl that the coefficients of the tube volume polynomial are invariants of the Riemannian metric. As a consequence, the intrinsic volumes are invariant under pullback along isometric immersions. The latter phenomenon, subsequently observed in a number of different geometric settings, is commonly referred to as the Weyl principle. In general normed spaces, the Holmes-Thompson intrinsic volumes naturally extend the Euclidean intrinsic volumes. After recalling the construction and the main properties of the Holmes-Thompson intrinsic volumes, we discuss to what extent the Weyl principle applies to Finsler manifolds. We show that while in general the Weyl principle fails, a weak form of the principle unexpectedly persists in certain settings. Based on joint work with D. Faifman.

Intrinsic volumes (called also Lipschitz-Killing curvatures) have been playing a key role in convex and integral geometry at least since Minkowski.In 1939 H. Weyl extended their definition to Riemannian manifolds. We formulate a few conjectures on their behavior under the Gromov-Hausdorffconvergence when the sectional curvature is bounded from below. We review some of the known special cases. Part of the conjectures is formulated in terms of a metric analogue of the nearby cycle construction from algebraic geometry.

Consider a sequence of n-dimensional closed Riemannian manifolds X_i with curvature uniformly bounded below converging in the Gromov-Hausdorff sense to an Alexandrov space X. Recently Semyon Alesker formulated several conjectures on the behavior of the intrinsic volumes of X_i. To this purpose he proposed a construction of an integer-valued function on X, which carries an additional geometric information about a subsequence of X_i. The definition of this function, however, is quite cumbersome, and it is not evident that the function is well-defined. In our work we confirm the existence of such functions in the situation when all X_i are closed surfaces. We describe completely all functions arising from this construction. The most interesting case is the case of collapse, i.e., when the dimension of X is less than the dimension of X_i.This is a joint work with Semyon Alesker and Mikhail Katz.

In this talk we will deal with Intrinsic Flat convergence defined by Sormani and Wenger using work of Ambrosio and Kirchheim. This is a generalization of Federer and Fleming Flat convergence for currents. We will show that given a closed and oriented manifold $M$ and Riemannian tensors $g_0 \leq g_j$ on $M$ that satisfy $vol(M, g_j)\to vol(M,g_0)$ and $diam(M,g_j)\leq D$ then $(M,g_j)$ converges to $(M,g_0)$ in the intrinsic flat sense. We note that under these conditions we do not necessarily obtain Gromov-Hausdorff convergence. We will show an analogous convergence result for manifolds with boundary. These results will be applied to show the stability of a class of tori with almost nonnegative scalar curvature and the stability of the positive mass theorem for a particular class of manifolds. [Based on joint work with Allen, Allen-Sormani, Cabrera Pacheco-Ketterer, Huang-Lee]

I am going to discuss some ways of using integral geometry, which I encountered in my work (with collaborators) in Riemannian and metric geometry, PDEs, and geometric control theory.  

In this talk we give a survey on the Laplacians on euclidean simplicial complexes and introduce a Laplacian on spherical simplicial complexes motivated by the theory of mixed volumes. Several results and a conjecture on the signature of this discrete spherical Laplacian will be stated.

I will survey results related to the following question: "Which finite metric spaces appear as subspaces of Alexandrov spaces with curvature bounded below or above?"

I will recall the standard problem of prescribing the Gauss curvature of a convex body in Euclidean space, and then introduce the hyperbolic case. In this new set-up, the curvature prescription problem has a unique solution, the uniqueness property follows from tools of integral geometry. This is joint work with P. Castillon

We establish an analogue of the classical Steiner formula in the $L_p$ Brunn Minkowski theory. The classical Steiner formula is a special case of this more general $L_p$ Steiner formula. We investigate the properties of the new $L_p$ Steiner coefficients and show, among other properties, that they are valuations. Based on joint works with Kateryna Tatarko.