gencurv2018 - epflsma.epfl.ch/~troyanov/booklet.pdfisomorphic as normed spaces, are generalized...

GenCurv2018

Generalized Curvatures and Related Topics

Summer School Bernoulli Center - EPFL03-07 September 2018

Organization

Maroussia Schaffner PortilloIvan IzmestievMarc Troyanov

Minicourses by Andreas Bernig (Frankfurt) Matthias Keller (Postdam) Marc Troyanov (EPFL) Thomas Richard (Paris-Est Créteil) Boris Springborn (TU Berlin)

Lectures byJérôme Bertrand (Toulouse)François Fillastre (Cergy-Pontoise)Udo Hertrich-Jeromin (TU Wien) Wai Yeung Lam (Brown) Mark Pauly (EPFL) Helmut Pottmann (TU Wien) Giona Veronelli (Paris XIII)

Timetable MONDAY, 09/03

8:30 - 9:15 Registration

9:15 - 9:30 Conference Opening

9:30 - 11:00 A. Bernig

11:00 - 11:30 Coffee Break

11:30 - 13:00 T. Richard

13:00 - 14:30 Lunch

14:30 - 16:00 M. Keller


16:30 - 17:30 U. Hertrich-Jeromin

17:45 - 19:00 Poster session and Wine & Cheese

TUESDAY, 09/04

9:00 - 10:30 A. Bernig

10:30- 11:00 Coffee Break

11:00 - 12:30 B. Springborn

12:30 - 13:45 Lunch

13:45 - 14:45 M. Troyanov


15:15- 16:15 M. Pauly

16:15- 17:15 H. Pottmann

WEDNESDAY, 09/05



11:00 - 12:30 M. Keller

12:45 - 20:00

Social Program : St Leonard Underground Lake, Visit to the "Cave Emery” wine vault, aperitif and wine tasting.

THURSDAY, 09/06

9:00 - 10:30 M. Troyanov


11:00 - 12:30 T. Richard

12:30 - 13:45 Lunch



15:15- 16:15 W. Y. Lam

16:15- 17:15 J. Bertrand

FRIDAY, 09/07

9:00 - 10:00 T. Richard

10:00- 11:00 A. Bernig


11:30 - 12:30 M. Keller

12:30 - 13:45 Lunch

13:45 - 14:45 M. Troyanov


15:15- 16:15 G. Veronelli

16:15- 17:15 F. Fillastre

Minicouses Andreas Bernig: Lipschitz-Killing Curvatures

Lecture 1: TUBE FORMULAS

• Steiner's formula• The volume of a euclidean tube around a convex body is a polynomial in the radius. The coefficients

are called intrinsic volumes. They have a nice explicit description if the convex body is either a polytope or has smooth boundary.

• Weyl's principle• The volume of a small euclidean tube around a compact submanifold in euclidean space is again a

polynomial. The coefficients are called Lipschitz-Killing curvatures. They can be expressed in terms of the intrinsic geometry of the submanifold (involving curvature terms such as scalar curvature).

Lecture 2: THEORY OF VALUATIONS

• Hadwiger's theorem and Crofton style kinematic formulas• The intrinsic volumes are valuations, i.e. finitely additive measures. Hadwiger gave a

characterization of them, which can be used to obtain alternative expressions and important geometric formulas.

• Valuations on manifolds• The Lipschitz-Killing curvatures induce a family of valuations and curvature measures on each

Riemannian manifold. In the case of subanalytic subsets, we describe the scalar curvature measure explicitly.

Lecture 3: MODERN THEORY OF VALUATIONS, and solutions to exercises.• Hermitian integral geometry• In hermitian spaces one can define a family of hermitian intrinsic volumes. They satisfy again some

kinematic formulas. Recently, kinematic formulas for complex projective spaces were shown.

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Matthias Keller: Upper curvature bounds and spectral theory.

We study a notion of curvature on planar graphs which goes back to ideas Descartes. Our main focus is put on geometric implications of upper bounds on the curvature. In turn these geometric implications relate to the spectrum of the graph Laplacian. In a further step we discuss how to implement these ideas to study higher dimensional objects such as polygonal complexes and buildings.

Thomas Richard: Intrinsic geometry of metric spaces with curvature bounded from below : convex polyhedra and beyond.

We will first observe that smooth surfaces with positive curvature and convex polyhedron have much in common in term of their intrinsic geometry. We will then abstract one of these propriety to give a definition of what it means for a geodesic metric space to have nonnegative curvature, a notion first introduce by Alexandrov and later revived by Burago-Gromov-Perelman. Using this notion we will give “metric proofs” of some well known theorems about positively curved manifolds.

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Boris Springborn: Discrete conformal maps, Uniformization and Hyperbolic Polyhedra

In recent years, a growing theory of discrete conformal maps has emerged, which is based on a simple notion of conformal equivalence for triangle meshes: Two triangle meshes are considered equivalent, if their edge lengths are related by scale factors associated to the vertices. This leads to a surprisingly rich theory with applications in computer graphics and geometry processing. On the purely mathematical side, the theory is intimately connected with hyperbolic geometry. Uniformization problems for discrete Riemann surfaces are equivalent to geometric realization problems for ideal hyperbolic polyhedra with prescribed intrinsic metric.

The purpose of these lectures is to present an introductory overview of this theory of discrete conformal maps and its connections to hyperbolic geometry.

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Marc Troyanov: Alexandrov Surfaces with Bounded Integral Curvature

Our goal in these lectures is to give an introduction to Alexandrov’s theory of surfaces with bounded integral curvature (BIC surfaces). There are several ways to characterise these surfaces and one of our aim is to relate the various viewpoints. We hope to cover the following topics:

• Curves in arbitrary metric spaces and their total curvature.• Sequence of metric spaces and their convergence.• Alexandrov’s BIC surfaces: definition and examples.• The curvature measure of BIC surfaces.• Convergence theorems.• Conformal structures and uniformisation of BIC surfaces.

Lectures Jérôme Bertrand : Alexandrov meets Kantorovitch.

In this talk, I will explain how some tools from optimal mass transport theory can be used to solve a curvature prescription problem for convex bodies. The problem concerns the Gauss curvature, considered in a generalised sense, of a general convex body. After a review of the method in the classical Euclidean setting, I will present a solution to this problem for convex bodies in the hyperbolic space. This is based on joint work with P. Castillon.

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François Fillastre: Hyperbolic geometry of shapes of convex bodies.

We define a distance on the space of convex bodies in the n-dimensional Euclidean space, up to translations and homotheties, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient Lorentzian structure is an extension of the intrinsic area form of convex bodies. We deduce that the space of shapes of convex bodies (i.e. convex bodies up to similarities) has a proper distance with curvature bounded from below by −1. In dimension 3, this space naturally identifies with the space of distances with non-negative curvature on the 2-sphere. Joint work with Clément Debin (SISSA, Trieste).

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Udo Hertrich-Jeromin: Obfuscating curvature

Curvature, a key invariant to describe geometric shape, a priori depends on a distance measurement. However, certain curvature conditions can be formulated in a way that only relies on a conformal or contact structure. We shall discuss a method to encode such curvature conditions that also lends itself easily to discretization.

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Wai Yeung Lam: Discrete minimal surfaces from quadratic differentials

There are various ways to define mean curvature for a polyhedral surface in space. One approach is via Steiner’s formula, which involves the area of its parallel surfaces induced by the Gauss map. However in literature, there are various definitions of Gauss map for polyhedral surfaces. Different choices lead to different formulas for mean curvature. We are interested in distinguishing those with rich mathematical structure and with connections to other discrete theories. We study a discrete analogue of minimal surfaces, which are surfaces with vanishing mean curvature. For a suitable notion of mean curvature, we establish connections between discrete minimal surfaces and discrete holomorphic functions like circle packings. It resembles the classical Weierstrass representation for minimal surfaces. Along the way, we introduce holomorphic quadratic differentials on graphs. They serve as a bridge between several discrete theories.

Mark Pauly: Mapping Materials - Complex 3D forms from 2D sheet materials.

In this talk I will outline how curvature theory can be leveraged to devise computational tools for the design of complex 3D surfaces that can be realized from planar sheet materials with unique physical properties. In particular, I will show how the results of Chebyshev inform algorithms to design with wiremesh materials and how conformal geometry provides effective tools for modeling with isotropic auxetic materials. Recent work studies how the desired 3D surface can be directly programmed into the 2D material, such that a simple deployment via inflation or gravitational loading automatically deforms the planar material towards its 3D target configuration. I’ll show that such structures can approximate target surfaces of positive mean curvature and bounded scale distortion relative to a given reference domain.

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Helmut Pottmann: Discrete geometric structures motivated by applications in architecture

We present recent research results and open problems on discrete structures which are motivated by architectural projects. A search for fair architectural skins from flat panels motivates the study of smoothness concepts for polyhedral surfaces. We present several approaches to “smooth” polyhedral surfaces, outline their use for a discrete curvature theory and discuss a relation to material minimizing structures. Curved facades from spherical panels and the design and construction of certain curved support structures with repetitive parameters lead to various discrete structures from circles and spheres. These include sphere geometric watertight extensions of principal meshes and quad nets with spherical vertex stars. The latter open up a new type of discrete surface parameterizations along curves of constant normal curvature and a simple way of controlling mean curvature in discrete structures.

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Giona Veronelli: Scalar curvature and metric spaces

We will consider the problem of finding a good notion of scalar curvature of a metric space. We are specially interested in the (a priori easy) case of Alexandrov spaces with lower bounds on the curvature. We will recall some of the few definitions proposed so far and we will focus on possible (dimensional) definitions based on the infinitesimal expansion of the volumes or of the q-extents of small geodesic balls.

Posters Affolter Niklas: Miquel dynamics, Clifford dynamics and the Dimer model.

A Z² circle pattern is a map from the Z² lattice to R² such that the vertices of each face of Z² are mapped to a circle. Using Miquel's six circle theorem, Ramassamy and Kenyon introduced discrete time dynamics that replace every second circle with a new circle, such that a Z² circle pattern is mapped to a new Z² circle pattern. We show that if we consider the circle centers, the time evolution of Miquel dynamics actually produces Clifford lattices. These octahedral lattices have been studied by Konopelchenko and Schief and consist locally of Clifford four circle configurations.The dimer model considers the set of all perfect matchings on a graph, and assigns each perfect matching a probability proportional to the product of the edge weights of the edges occuring in that matching. We show that if we use the distances between circle centers as edge weights of a dimer model, Miquel dynamics preserve probabilities in what is known as urban renewal.

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Bartelmes Nina: Monochromatic metrics are generalized Berwald.

I will show that monochromatic Finsler metrics, i.e., Finsler metrics such that each two tangent spaces are isomorphic as normed spaces, are generalized Berwald metrics, i.e., there exists an affine connection, possibly with torsion, that preserves the Finsler function. Furthermore I will present some global results on the existence of generalized Berwald metrics and give an outlook on the Landsberg unicorn problem.

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Buro Guillaume: What makes the Devil’s staircase so devilish?

We discuss under what hypothesis one can compute the length of a rectifiable curve in an arbitrary metric space by integrating the "speed" of the curve. A fundamental counter-example is given by the "Devil’s staircase" (Cantor-Vitali function), which is a rectifiable curve that is not absolutely continuous. It is standard and easy to prove that this curve is not absolutely continuous. The Banach-Zarecki Theorem gives a deeper explanation of this phenomenon in terms of measure theory. We will state that theorem and give the main ideas of its proof.We will also discuss how to actually compute the length of the Devil’s staircase.

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Gutiérrez-Rodríguez Ixchel: Bach-flat gradient Ricci solitons on affine surfaces.

The classification of Bach-flat gradient Ricci solitons is a problem of special interest. Our purpose is to construct new examples of Bach-flat gradient Ricci solitons in the neutral signature case which are neither half conformally flat nor conformally Einstein and generalize our construction to characterize Bach-flat Riemannian extensions of affine surfaces admitting a nilpotent structure.

Kamtue Supanat: Rigidity for the discrete Bonnet-Myers diameter bound. Which graphs look like a sphere?

Bonnet-Myers theorem gives an estimate of the diameter in terms of a positive Ricci curvature bound of a manifold. In the discrete setting of graphs, Ollivier’s notion of Ricci curvature provides a discrete analogue of the Bonnet-Myers theorem. In view of Cheng’s rigidity result, we classify all graphs for which the Bonnet-Myers estimate is sharp, under the extra assumption of self-centeredness (i.e. each vertex has another vertex whose distance between them is equal to the diameter). A complete classification of all self-centered Bonnet-Myers sharp graphs is given by hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs J(2n, n), the Gosset graph and suitable Cartesian products.

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Labeni Hicham: Isometric immersion in Anti-de-Sitter space.

We will present the main steps, to prove the existence of an isometric immersion of a metric space (S,d) with curvature ≤ -1 (in the sense of Alexandrov ) in Anti de-Sitter space, under some conditions implying convexity, when we impose that the associated representations leave a plan invariant.

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Lang Julius: Smoothness of projective transformations in dimension 2.

A path geometry is a system of unparametrized curves where for every point and every tangent direction there is exactly one such curve – the typical example are geodesics of a Finsler or Riemannian metric. We present a proof that every continuous injective map, taking a smooth path geometry in the plane projectively to a smooth path geometry, is smooth itself. This extends a result by Frederick Brickell from 1965 valid for dimension three and higher

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Marcone Adrien: Gauss-Bonnet formula for manifolds with conical ends.

The celebrated Gauss-Bonnet-Chern theorem holds for compact Riemannian manifolds and relates the Euler-characteristic 𝜒(M) of the manifold to its total curvature ∫M Pf(Ω). A strong assumption on the geometry of the manifold allows us to drop the compactness hypothesis and consider manifolds with conical ends. We compute the difference ∫M Pf(Ω) - 𝜒(M) in terms of the Lipschitz-Killing curvatures of some compact manifolds generating the conical ends. The proof is completely intrinsic as the manifold is not assumed to be embedded in some euclidean space.

Rasskin Ivan: On the tangency graphs of ball-packings and linkless graphs.

A ball-packing is a family of n-balls (of different size) in Rn compactified with pairwise disjoint interiors. The tangency graph of a ball-packing is the graph where the vertices are the set of balls and there is an edge between two vertices if the corresponding balls are tangent. For n=1 the tangency graphs are the paths and cycles. For n=2 we know by the Koebe-Andreev-Thurston's theorem that the tangency graphs are the planar graphs. For n=3 there is no full characterisation. On the other hand, two closed curves C1 and C2 in R3 are linked if every borded surface with boundary C1 intersects C2. A graph is linkless if there is an embedding of the graph in R3 such that every two cycles of the graph are not linked. Using the relation between the curvatures of the balls we show that the family of linkless graphs and the tangency balls of 3-balls are incomparable by inclusion.

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Sidler Hubert: Harmonic quasi-isometric maps into Gromov hyperbolic CAT(0)-spaces.

The poster is about the recent joint work with Stefan Wenger, where we proved the following theorem, which generalizes recent results of Benoist-Hulin and Markovic. Let X be a Hadamard manifold with negatively pinched sectional curvature, and Y be a Gromov hyperbolic, locally compact, CAT(0)-space. If f : X —> Y is a quasi-isometric map, then there exists an energy-minimizing harmonic map h:X -> Y within bounded distance of f. Besides this statement, the most important definitions and the outline of its proof, the poster also refers to previous and related results.

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Sukhorebska Darya: Simple close geodesics on regular tetrahedra in hyperbolic space.

We explore closed geodesies on the regular tetrahedra in hyperbolic 3-space. Properties of closed geodesics on the regular tetrahedron in the hyperbolic space differ from one in Euclidean space. We present two necessary conditions that simple close geodesics on regular tetrahedra in the 3-dimensional hyperbolic space satisfy: such geodesics pass through the middle of two pairs of opposing edges, and can’t pass close to the vertices of the tetrahedron. Furthermore, we evaluated from above the number of simple close geodesics on regular tetrahedra in the hyperbolic space. Also, we explicitly describe one type of simple closed geodesics on regular tetrahedra in hyperbolic 3-space.

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Szewieczek Gudrun : Channel surfaces – smooth & discrete.

Smooth channel surfaces, envelopes of a 1-parameter family of spheres, were studied by classical geometers as Monge and Blaschke using different geometric data. In particular, these surfaces are characterised by the existence of one family of circular curvature lines, hence one of the principal curvatures is constant along its curvature directions. We discuss how we can formulate these properties for discrete nets in Lie sphere geometry, so-called discrete Legendre maps. In this way we obtain various geometric data living on vertices, edges and faces of the net, which describes discrete channel surfaces

List of participants Affolter Niklas, TU BerlinAlabdulsada Layth M, U. of DebrecenBartelmess Nina, JenaBernig Andreas, FrankfurtBertrand Jérôme, ToulouseBuro Guillaume, EPFLColbois Bruno, NeuchâtelColombo Maria, ETHZ and EPFLCorro Diego, KarlsruheCreutz Paul, Universität KölnDe Rosa Luigi, EPFLEbli Stefania, EPFLFillastre François, Cergy-PontoiseFischer Florian, PotsdamFitzi Martin, FribourgGarin Adélie, EPFLGasparetto Carlo, TriesteGuo Changyu, EPFLGutiérrez-Rodríguez Ixchel, Santiago de CompostelaHacker Celia, EPFLHeer Loreno, Universität ZürichHertrich-Jeromin Udo, TU WienIvanov Grigory, EPFLIzmestiev Ivan, FribourgKamtue Supanat, Durham UniversityKeller Matthias, PotsdamKourimska Hana, TU BerlinLabeni Hicham, Cergy-PontoiseLam Wai Yeung, Brown UniversityLang Julius, JenaMarcone Adrien, EPFLMerlin Louis, University of LuxembourgPauly Mark, EPFLParise Davide, EPFL and CambridgePerrin Hélène, FribourgPottmann Helmut, TU WienProsanov Roman, FribourgRasskin Ivan, MontpellierRichard Thomas, Paris-Est CréteilSemmler Klaus-Dieter, EPFL

Sidler Hubert, Université de FribourgSivak Iryna, EPFLSpreemann Gard, EPFLSpringborn Boris, TU BerlinSukhorebska Darya, KharkivSzewieczek Gudrun, TU WienTroyanov Marc, EPFLVeronelli Giona, Paris XIII

gencurv2018 - epflsma.epfl.ch/~troyanov/booklet.pdfisomorphic as normed spaces, are generalized...

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