Pangolin Seminar
Zoom seminar organized by Sébastien Alvarez, François Fillastre, Andrea Seppi and Graham Smith
Every other Tuesday
09h00 New York, 14h00 London, 10h00 Montevideo, 15h00 Paris, 10h00 Rio de Janeiro
The Zoom link will be posted here 15 minutes before the start of the lecture.
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Hessian estimates for the special Lagrangian equation by doubling
The special Lagrangian equation is a fully nonlinear elliptic PDE whose solutions correspond to minimal graphs of the form (x,u’(x)). The interior estimates by Chen, Wang, Warren, and Yuan a decade ago depend heavily on the theory of minimal surfaces. In this talk, we describe a new proof of these estimates using a doubling approach. This compactness method combines the maximum principle, Savin’s small perturbation theorem, and Chaudhuri-Trudinger’s Alexandrov theorem. This talk will survey the PDE and the principles behind the method.
08 December 2021 - 31 January 2022
Summer Recess
Strict hyperbolization and special cubulation
Gromov introduced some “hyperbolization” procedures to turn a given polyhedron into a space of non-positive curvature, in a way that preserves some of the topological features of the original polyhedron. For instance, a manifold is turned into a manifold. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. This has been used to construct examples of manifolds and groups that exhibit various pathologies, despite having negative curvature. We will describe these procedures, and how to construct geometric actions of the resulting groups on CAT(0) cube complexes. As an application, we find that many hyperbolic groups resulting from strict hyperbolization are virtually linear over the integers. This is joint work with J. Lafont.
Translating solitons of the mean curvature flow in cohomogeneity one manifolds
We study new examples of translating solitons of the mean curvature flow. We consider for this purpose manifolds admitting pseudo-Riemannian submersions and cohomogeneity one actions by isometries on suitable open subsets. This general setting also covers the well-known classical Euclidean examples of translating solitons invariant by some group actions. As an application, we completely classify the rotationally invariant translating solitons in Minkowski space.
The homotopy type of spaces of locally convex curves in the sphere
A smooth curve $\gamma: [0,1] \to S^2$ is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally positive curves $\gamma$ with $\gamma(0) = \gamma(1) = e_1$ and $\gamma'(0) = \gamma'(1) = e_2$ has three connected components. Our first aim is to describe the homotopy type of these spaces. One of the connected components is known to be contractible. The two other connected components are homotopically equivalent to $(\Omega S^3) \vee S^2 \vee S^6 \vee S^{10} \vee \cdots$ and $(\Omega S^3) \vee S^4 \vee S^8 \vee S^{12} \vee \cdots$, respectively: we describe the equivalence.
More generally, a smooth curve $\gamma: [0,1] \to S^n$ is locally convex if \[ \det(\gamma(t), \gamma'(t), \ldots, \gamma^{(n)}(t)) > 0 \] for all $t$. A motivation for considering this space comes from linear ordinary differential equations. Again, we would like to know the homotopy type of the space of locally convex curves with prescribed initial and final jets. We present several partial results.
Includes joint work with E. Alves, V. Goulart, B. Shapiro, M. Shapiro, C. Zhou and P. Zuhlke.
Shortest closed curve to inspect a sphere
We show that in Euclidean 3-space any closed curve which lies outside the unit sphere and contains the sphere within its convex hull has length at least 4Pi. Equality holds only when the curve is composed of 4 semicircles of length Pi, arranged in the shape of a baseball seam, as conjectured by V. A. Zalgaller in 1996. This is joint work with James Wenk.
Effective high-temperature estimates ensuring a spectral gap
The main goal of the talk shall be to explain a few ideas from two classical theories : the thermodynamical formalism, and the perturbation of linear operators. The "thermodynamical formalism" is a framework to describe particular invariant measures of dynamical systems, called "equilibrium states", parametrized by functions on the phase space, called "potentials". This formalism is based on the "transfer operator"; when this operator has a spectral gap, the equilibrium state exists, is unique, and has very good statistical properties (exponential mixing, Central Limit Theorem, etc.) If one perturbs slightly the potential, the corresponding transfer operator is also perturbed.
The classical theory of perturbation of operators ensures that the spectral gap property is an open condition and that under bounded pertubration, the eigendata of an operator depends analytically on the perturbation. It turns out that using the Implicit Function Theorem, this theory can be made effective with explicit bounds on the size of a neighborhood where the spectral gap persists. Using this effective perturbation theory, we show completely explicit bound on the potential ensuring the spectral gap property for transfert operators of classical families of dynamical systems.
How many Anosov flows can you put on a (closed, hyperbolic) 3 manifold?
This question is one part of the puzzle connecting the topology and geometry of a manifold to the possible dynamical systems that it supports, in this case the classification problem of Anosov flows. I will motivate this question and describe some work with Jonathan Bowden constructing flows on hyperbolic 3-manifolds, as well as some recent joint work on the classification problem joint with Thomas Barthelme and Steven Frankel.
Finiteness Theorems for Gromov-Hyperbolic Groups
This is a joint work with G. Courtois, S. Gallot and A. Sambusetti. We shall prove that, given two positive numbers $\delta$ and $H$, there are finitely non cyclic torsion-free $\delta$-hyperbolic marked group $(\Gamma , \Sigma)$ satisfying ${\rm Ent} (\Gamma , \Sigma) \le H$, up to isometry (of marked groups). Here a marked group is a group $\Gamma$ together with a symmetric generating set $\Sigma$ and ${\rm Ent}$ is the entropy of the marked group. These notions will be defined precisely.
14 July 2021 - 13 September 2021
Winter Recess
FC-subspaces
In a joint work with Nikolay Bogachev, Alexander Kolpakov, and Leone Slavich we discovered an interesting connection between totally geodesic subspaces of a hyperbolic manifold or orbifold and finite subgroups of the commensurator of its fundamental group. We call the totally geodesic subspaces associated to the finite subgroups by fc-subspaces. It appears that these subspaces have some remarkable properties. We show that in an arithmetic orbifold all totally geodesic subspaces are fc and there are infinitely many of them, while in non-arithmetic cases there are only finitely many fc-subspaces and their number is bounded in terms of volume. In the talk, I will discuss these results and if time permits will sketch some other applications of fc-subspaces.
Sharp Ellipsoid Embeddings and Toric Mutations
We will show how to construct volume filling ellipsoid embeddings in some 4-dimensional toric domain using mutation of almost toric compactification of those. In particular we recover the results of McDuff-Schlenk for the ball, Fenkel-Müller for the product of symplectic disks and Cristofaro-Gardiner for E(2,3), giving a more explicit geometric perspective for these results. To be able to represent certain divisors, we develop the idea of symplectic tropical curves in almost toric fibrations, inspired by Mikhalkin's work for tropical curves. This is joint work with Roger Casals.
Obs: The same result appears in "On infinite staircases in toric symplectic four-manifolds", by Cristofaro-Gardiner -- Holm -- Mandini -- Pires. Both papers were posted simultaneously on arXiv.
Lojasiewicz inequalities near simple bubble trees
In the study of (almost-)critical points of an energy functional one is often confronted with the problem that a weakly-obtained limiting object does not have the same topology. For example sequences of almost-harmonic maps from a surface will in general not converge to a single harmonic map but rather to a whole bubble tree of harmonic maps, which cannot be viewed as an object defined on the original domain.
One of the consequences of this phenomenon is that one of the most powerful tools in the study of (almost-)critical points and gradient flows of analytic functionals, so called Lojasiewicz-Simon inequalities, no longer apply.
In this talk we discuss a method that allows us to prove such Lojasiewicz inequalities for the harmonic map energy near simple trees and explain how these inequalities allow us to prove convergence of solutions of the corresponding gradient flow despite them forming a singularity at infinity.
Singular metrics of constant non-local curvature
We will consider the problem of constructing singular metrics of constant non-local curvature in conformal geometry, using a gluing scheme. This non-local curvature is defined from the conformal fractional Laplacian, a Paneitz type operator of non-integer order. For the gluing process, one needs a model solution which is given by the solution of a non-local ODE with good conformal properties. It turns out that conformal geometry provides powerful tools for the analysis of such equations.
The topology of constant mean curvature surfaces with convex boundary
We discuss old and new results about the shape of a constant mean curvature surface with strictly convex boundary, contained in the halfspace determined by the surface containing the boundary.
On tilings, amenable equivalence relations and foliated spaces
I will describe a family of foliated spaces constructed from tylings on Lie groups. They provide a negative answer to the following question by G.Hector: are leaves of a compact foliated space always quasi-isometric to Cayley graphs? Their construction was motivated by a profound conjecture of Giordano, Putnam and Skau on the classification, up to orbit equivalence, of actions of countable amenable groups on the Cantor set. I will briefly explain how these examples relate to the GPS conjecture. This is joint work with Fernando Alcalde Cuesta and Álvaro Lozano Rojo.
The Weyl problem for unbounded convex surfaces in H^3
The classical Weyl problem in Euclidean space, solved in the 1950s, askswhether any smooth metric of positive curvature on the sphere can be realized as the induced metric on the boundary of a unique convex subset in $\R^3$. It was extended by Alexandrov to the hyperbolic space, where a dual problem can also be considered: prescribing the third fundamental form of a convex surface.
We will describe extensions of the Weyl problem and its dual to unbounded convex surfaces in $H^3$. Two types of generalizations can be stated, one concerning unbounded convex surfaces, the other unbounded locally convex surfaces. Both questions have as special cases a number of known result or conjectures concerning 3-dimensional hyperbolic geometry, circle packings, etc. We will present a rather general existence result concerning convex subsets.
Growth of quadratic forms under Anosov subgroups
For positive integers p and q we define a counting problem in the (pseudo-Riemannian symmetric) space of quadratic forms of signature (p,q) on R^{p+q}. This is done by associating to each quadratic form a geodesic copy of the Riemannian symmetric space of PSO(p,q) inside the Riemannian symmetric space of PSL_{p+q}(R), and by looking at the orbit of this geodesic copy under the action of a discrete subgroup of PSL_{p+q}(R). We then present some contributions to the study of this counting problem for Borel-Anosov subgroups of PSL_{p+q}(R).
On the relations between the universal Teichmuller space and Anti de Sitter geometry
Anti de Sitter (AdS) space is the Lorentzian cousin of the hyperbolic 3-space: it is a symmetric space with constant curvature -1. In this talk, we will consider surface group representations in the isometry group of AdS space, called quasi-Fuchsian representations. There is 2 classical objects associated to those representations and one of the goal is to understand their interplay: the limit set which is a quasi-circle in the boundary at infinity of AdS space and a convex set inside AdS which is preserved by the group action and bounded by two pleated surfaces. I will conclude the talk by a report on a work in common with Jean-Marc Schlenker where we extend the "Teichmüller" situation to the "universal Teichmüller".
Amenability of covers through critical exponents
Let M be a negatively curved manifold. If M is “strongly positively recurrent”, i.e. there is a critical gap between its entropy at infinity and its entropy, we show that a cover M' of M is amenable if and only if the critical exponents of M' and M coincide. The proof uses a construction of Patterson-Sullivan measures twisted by a representation. This is a joint work with R. Dougall, R. Coulon and S.Tapie.
Elliptic Linear Weingarten graphs with isolated singularities
We study isolated singularities of graphs whose mean and Gaussian curvature satisfy the elliptic linear relation $2\alpha H+\beta K=1$, $\alpha^2+\beta>0$. This family of surfaces includes convex and non-convex singular surfaces and also cusp-type surfaces. We determine in which cases the singularity is removable, and classify non-extendable totally isolated singularities in term of regular real analytic strictly convex curves in $\S^2$. This is a joint work with João P. dos Santos.
Families of equivariant immersions in $H^3$ with holomorphic holonomy
Given an equivariant immersion of a surface in the hyperbolic 3-space, a typical problem consists in understanding whether a deformation of the immersion (parametrized over a complex manifold) produces a holomorphic deformation of its mondromy in PSL(2,C). In this talk we present a simple “trick” providing a sufficient condition for this property, offering for instance an alternative proof of the holomorphicity of the complex landslide flow. This result is a consequence of the study of immersions into the space of geodesics of the hyperbolic 3-space, seen as a holomorphic Riemannian manifold, whose key features will be discussed in the talk.
This is joint work with Francesco Bonsante.
Deformed G2 instantons
Deformed G2-instantons are special connections occurring in G2 geometry in 7 dimensions. They arise as “mirrors” to certain calibrated cycles, providing an analogue to deformed Hermitian-Yang-Mills connections, and are critical points of a Chern-Simons-type functional. I will describe an elementary construction of the first non-trivial examples of deformed G2-instantons, and their relation to 3-Sasakian geometry, nearly parallel G2-structures, isometric G2-structures, obstructions in deformation theory, the topology of the moduli space, and the Chern-Simons-type functional.
Summer Recess - 16/12/2020 to 25/01/2021
Minimal Surfaces in Negatively Curved 3-Manifolds and Dynamics
The Grassmann bundle of tangent 2-planes over a closed hyperbolic 3-manifold M has a natural foliation by (lifts by their tangent planes of) immersed totally geodesic planes in M. I am going to talk about work I’ve done on constructing foliations whose leaves are (lifts of) minimal surfaces in a metric on M of negative sectional curvature, which are deformations of the totally geodesic foliation described above. These foliations make it possible to use homogeneous dynamics to study how closed minimal surfaces in variable negative curvature are distributed in the ambient 3-manifold. Many of the ideas here come from recent work of Calegari-Marques-Neves, which I will also talk about. I was able to establish some preliminary facts about the dynamics of these foliations, but much remains to be understood.
Sildes of this talk are available here.
Plateau problem in the pseudo-hyperbolic space
The pseudo-hyperbolic space H^{2,n} is the pseudo-Riemannian analogue of the classical hyperbolic space. In this talk, I will explain how to solve an asymptotic Plateau problem in this space: given a topological circle in the boundary at infinity of H^{2,n}, we construct a unique complete maximal surface bounded by this circle. This construction relies on Gromov’s theory of pseudo-holomorphic curves. This is a joint work with François Labourie and Mike Wolf.
Variation of Hodge structures and enumerating triangulations and quadrangulations of surfaces
Constant Gaussian curvature surfaces in hyperbolic 3-manifolds
In this talk I will describe how constant Gaussian curvature (CGC) surfaces interpolate the structures of the pleated boundary of the convex core and of the boundary at infinity of a geometrically finite hyperbolic end, and I will present a series of consequences of this phenomenon: a description of the renormalized volume of a quasi-Fuchsian manifold in terms of its CGC-foliation, a characterization of the immersion data of CGC-surfaces of a hyperbolic end as an integral curve of a time-dependent Hamiltonian vector field on the cotangent space to Teichmüller space, and a consequent generalization of McMullen’s Kleinian reciprocity theorem.
Glueing constructions of Compact Einstein four-manifolds with negative sectional curvature
We construct examples of closed Einstein four-manifolds with negative sectional curvature. We describe two main families of examples which are respectively obtained as ramified covers and smooth quotients of "large" hyperbolic 4-manifolds with symmetries. The first family of examples is sometimes referred to as Gromov-Thurston manifolds. The Einstein metrics that we construct are the result of a glueing procedure. They are obtained as deformations of an approximate Einstein metric which is an interpolation between a "black-hole – type" Riemannian Einstein metric near the symmetry locus and the hyperbolic metric. This construction yields the first example of compact Einstein manifolds with negative sectional curvature which are not locally homogeneous. This is a joint work with J. Fine (ULB, Brussels).
Geometric semi-norms in homology
The simplicial volume of a simplicial complex is a topological invariant related to the growth of the fundamental group, which gives rise to a semi-norm in homology. In this talk, we introduce the volume entropy semi-norm, which is also related to the growth of the fundamental group of simplicial complexes and shares functorial properties with the simplicial volume. Answering a question of Gromov, we prove that the volume entropy semi-norm is equivalent to the simplicial volume semi-norm in every dimension. Joint work with I. Babenko.
Marked length spectrum, geodesic stretch and pressure metric
The marked length spectrum of a negatively-curved manifold is the data of all the lengths of closed geodesics, marked by the free homotopy of the manifold. The marked length spectrum conjecture (also known as the Burns-Katok conjecture, 1985) asserts that two negatively-curved manifolds with same marked length spectrum should be isometric. This conjecture was proved on surfaces (Croke '90, Otal '90) but remains open in higher dimensions. I will present a proof of a local version of this conjecture, based on the notions of geodesic stretch and pressure metric (a generalization of the Weil-Petersson metric to the context of variable curvature). Some elements of the theory of Pollicott-Ruelle resonances and anisotropic spaces will also be needed (I will recall everything). Joint work with C. Guillarmou and G. Knieper.
On the existence of Killing fields in smooth spacetimes with a compact Cauchy horizon
We prove that the surface gravity of a compact non-degenerate Cauchy horizon in a smooth vacuum spacetime, can be normalized to a non-zero constant. This result, combined with a recent result by Oliver Petersen and István Rácz, end up proving the Isenberg-Moncrief conjecture on the existence of Killing fields, in the smooth differentiability class. The well known corollary of this, in accordance with the strong cosmic censorship conjecture, is that the presence of compact Cauchy horizons is a non-generic phenomenon.
Though we work in 3 + 1, the result is valid line by line in any n +1-dimensions.
The index theorem for manifolds with cusps.
I will speak on the result obtained with W. Ballmann and J. Brüning about index theorem on manifold with cusps :
-Eigenvalues and holonomy. Int. Math. Res. Not. 2003, no. 12, 657–665.
-Regularity and index theory for Dirac-Schrödinger systems with Lipschitz coefficients.
J. Math. Pures Appl. (9) 89 (2008), no. 5, 429–476.
-Index theorems on manifolds with straight ends. Compos. Math. 148 (2012), no. 6, 1897–1968.
I will start with a review of the case of compact manifolds and manifolds with cylindrical ends (i.e. the work of Atiyah-Patodi-Singer) and then describe the main technical difficulties we had to face.
11 August 2020 - Recess
Intrinsic volumes of submanifolds of normed spaces: How intrinsic are they?
The intrinsic volumes, or quermassintegrals, are certain geometric functionals on sufficiently nice subsets of Euclidean space, given by the coefficients of the volume of an epsilon-tube of the set, which is a polynomial in epsilon. H. Weyl discovered that their value on a Riemannian submanifold of Euclidean space is, remarkably, an intrinsic invariant of the metric. We will consider the setting of a normed space, where the Holmes-Thompson intrinsic volumes are available, and attempt to extend Weyl's result to Finsler submanifolds.
Based on a joint work with T. Wannerer.
The Gauss map for nearly-Fuchsian manifolds
In this talk we will study the Gauss map for nearly-Fuchsian manifolds, namely complete hyperbolic n-manifolds homeomorphic to HxR, where H is a closed hypersurface with principal curvatures smaller than one in absolute value. The Gauss map of such a hypersurface is a Lagrangian equivariant embedding in the space of oriented geodesics of hyperbolic space, which is known to have a natural para-Kähler structure. We will present two characterizations of the Lagrangian equivariant embeddings obtained in this way, the first in terms of the vanishing of the Maslov class, and the second in terms of orbits of the group of Hamiltonian symplectomorphisms.
This is joint work with Christian El Emam (Pavia).
On eternal forced mean curvature flows of tori in perturbations of the unit sphere
Using a singular perturbation argument based on the work of B. White, we construct eternal mean curvature flows of tori in perturbations of the standard unit 3-sphere. Besides being of interest in the theory of mean curvature flows, such objects have applications in Morse homology theory. A large part of the proof involves the construction of certain types of functions of Morse-Smale type over the moduli space of Clifford tori. This has interesting potential applications to the theory of Radon transformations. This is joint work with Claudia Salas Mangaño. https://arxiv.org/abs/2004.00054
16 June 2020 - Gabriel Calsamiglia, Universidade Federal Fluminense
Spaces of isoperiodic holomorphic and meromorphic differentials
I will present some results on the topology of the spaces of holomorphic and meromorphic one forms over complex curves for which integration along certain homology classes is constant.
Rigidity of compact Fuchsian manifolds with convex boundary
By a compact Fuchsian manifold with boundary we mean a hyperbolic 3-manifold homeomorphic to $S_g \times [0; 1]$ such that the boundary component $S_g \times \{ 0\}$ is geodesic. Here $S_g$ is a closed oriented surface of genus $g>1$. Fuchsian manifolds are known as toy cases in the study of geometry of hyperbolic 3-manifolds with boundary. In my talk I will sketch a proof that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced path metric on $S_g \times \{1\}$. We do not put further restrictions on the boundary except convexity. This unifies two previously known results: in the case of smooth boundary such a result follows from a work of Schlenker and in the case of polyhedral boundary it was proven by Fillastre.
Critical exponents in higher rank symmetric spaces
The aim of the talk is to present some recent results on critical exponents for discrete subgroups of higher rank semisimple Lie groups. We will survey classical results in negative curvature, the relationship with entropy and the Hausdorff dimension of limit sets. Then we will introduce the geometric properties of higher rank symmetric spaces and explain the main differences with strict negative curvature. We will focus on two different results : the behaviour of the critical exponent under normal subgroup (j.w. S. Tapie) and the extension of classical results to pseudo-riemannian geometry (j.w. D. Monclair).
Earthquakes and graftings of hyperbolic surface laminations
We study compact hyperbolic surface laminations. These are a generalization of closed hyperbolic surfaces which appear to be more suited to the study of Teichmüller theory than arbitrary non-compact surfaces. We show that the Teichmüller space of any non-trivial hyperbolic surface lamination is infinite dimensional. In order to prove this result, we study the theory of deformations of hyperbolic surfaces, and we derive what we believe to be a new formula for the derivative of the length of a simple closed geodesic with respect to the action of grafting. This formula complements those derived by McMullen in [23], in terms of the Weil-Petersson metric, and by Wolpert in [33], for the case of earthquakes. https://arxiv.org/abs/1907.12126
On the asymptotic structure of finite-type k-surfaces in 3-dimensional hyperbolic space
For k∈]0,1[ a finite-type k-surface in 3-dimensional hyperbolic space is defined to be a complete, immersed surface of finite area and of constant extrinsic curvature equal to k. In [25] we showed that the space Sk of finite-type k-surfaces in H3 is homeomorphic to the space of pointed ramified coverings of the extended complex plane C^. Every finite-type k-surface (S,e) has finitely many ends, each of which is asymptotic to an immersed cylinder wrapping finitely many times, ever more tightly, about a complete geodesic ray. We show that each end of (S,e) wraps around a preferred geodesic, defined in terms of Steiner curvature centroids, which we call the Steiner geodesic. Whilst one extremity of each Steiner geodesic coincides with the extremity of its end, the other defines another point of C^ which we call the Steiner point of that end. We derive algebraic relations satisfied by the Steiner points of a finite-type k-surface. Finally, we introduce the generalised volume and renormalised energy of finite-type k-surfaces as functions over Sk and we prove a Schläfli type formula relating their derivatives to the Steiner points. We conclude, in particular, that, when considered as observable quantities over Sk, the extremity and Steiner point of each end together constitute a pair of conjugate variables over this space. https://arxiv.org/abs/1908.04834