IIT Bombay Geometric Analysis Seminar

Abstract: Given a closed manifold $(M^n,g)$, $n\geq 3$, Olivier Druet proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation $$\Delta_g u+h_0u=u^{\frac{n+2}{n-2}},\u>0\hbox{ in }M$$ is that $h_0\in C^1(M)$ touches the Scalar curvature somewhere when $n\geq 4$ (the condition is different for $n=6$). In this paper, we prove that Druet's condition is also sufficient provided we add its natural differentiable version. For $n\geq6$, our arguments are local. For the low dimensions $n\in\{4,5\}$, our proof requires the introduction of a suitable mass that is defined only where Druet's condition holds. This mass carries global information both on $h_0$ and $(M,g)$. Joint work with J\'er\^ome V\'etois (McGill University, Canada).

Abstract: Flows of frames over negatively curved Riemannian manifolds (M, g) are one of the oldest examples of partially hyperbolic dynamics. It is well known that frame flows of hyperbolic manifolds are ergodic, while Kahler manifolds never have ergodic frame flows; Brin conjectured in the 70's that all manifolds with sectional curvature between -1 and -0.25 (i.e. curvature is 0.25-pinched) have ergodic frame flows. In this talk I will explain recent progress on this conjecture: we show that in dimensions 4k+2 the frame flow is ergodic if (M, g) is ~0.27 pinched, and in dimensions 4k if it is ~0.55 pinched. Our new method uses techniques in hyperbolic dynamics (transitivity group, Parry's representation), topology of structure groups of spheres, and Fourier analysis in the vertical fibre of the unit sphere bundle (based on Pestov identity). This is joint work with Lefeuvre, Moroianu, and Semmelmann.

Abstract: Studying the concentration of eigenfunctions of Schrödinger or Laplace-type operators is a well known problem of semiclassical analysis. Useful tools for the latter include semiclassical measures, WKB expansion. After recalling some classical results of this theory we will focus on less classical ones that are related to the study of analytic eigenbranches or eigenvalue spacing problems. Based on joint work with C. Judge and J. Marzuola.

Abstract: In this talk we consider sign-changing solutions of critical Yamabe-Schrodinger type equations of second order. Unlike their positive counterpart these solutions have no direct physical or geometrical meaning, but have been shown to arise in geometrical contexts. They appear for instance as extremals for higher eigenvalues minimisation problems in a given conformal class. We describe in this talk the structure of bubbling sign-changing solutions for these equations and provide a detailed asymptotic description, in strong spaces, of the blow-up. As a consequence we prove some precompactness results for the set of energy-bounded solutions of these equations. Some of these results have been obtained in collaboration with J. Vétois (McGill University).

Abstract: On a bounded domain of the Euclidean space $\mathbb{R}^{2m}$, m>1, Adimurthi, Robert and Struwe pointed out that, even assuming a volume bound $\int e^{2mu} dx\le C$, some blow-up solutions for prescribed Q-curvature equations $(-\Delta)^m u= Q e^{2m u}$ without boundary conditions may blow-up not only at points, but also on the zero set of some nonpositive nontrivial polyharmonic function. This is in striking contrast with the two dimensional case (m=1). During this talk, we will discuss the construction of such solutions which involve (possible generalizations of) the Walsh-Lebesgue theorem and some issues about elliptic problems with measure data.

Abstract: Let L(G) denotes the Laplacian of a graph G. The second smallest eigenvalue of L(G) is called the spectral gap. In this talk, we discuss two spectral gap bounds for the reduced Laplacian of a simplicial complex. As an application we prove that, if the spectral gap of the Laplacian of skeleton of a simplicial complex is enough large, then its co-homology vanishes up to certain dimensions. We also see an application of these results in certain random complexes. This is a joint work with D. Yogeshwaran.

Abstract: Resonances of Riemannian manifolds are often studied with tools of microlocal analysis. I will discuss some recent results on upper fractal Weyl bounds for certain hyperbolic surfaces of infinite area, obtained with transfer operator techniques, which are tools complementary to microlocal analysis. This is joint work with F. Naud and L. Soares.

Abstract: Let $\Omega\subset\mathbb{C}$ be a bounded domain. The localization landscape (a.k.a torsion function) of $\Omega$ is a function v which satisfies $\Delta v = - 2$ in $\Omega$ , with boundary data $v(z) =0$, for $z\in\partial\Omega$. In this talk, we will present an upper bound for the number of critical points of v in various domains where we can make sense of some notion of "order" of the domain. We will talk about connections to eigenvalue problems and conclude the talk with some open problems. Based on joint work with Erik Lundberg.

Abstract: Due to the presence of the exponential nonlinearity, the Liouville equation in dimension three and higher is supercritical. In particular, it admits several singular solutions. We will talk about asymptotic behaviour of a family of stationary solutions, and how to use it to obtain partial regularity results.

Abstract: I will discuss convex sets in Heisenberg group and give a classification result for geodetically convex sets.

Abstract: Given a 2-dimensional closed surface, we will show that the Moser-Trudinger functional has critical points of arbitrarily high energy. Since the functional is too critical to directly apply to it the known variational methods (in particular the Struwe monotonicity trick), we will approximate it by subcritical ones, which in fact interpolate it to a Liouville-type functional from conformal geometry. Hence our result will also unify and give common results for these two apparently unrelated problems. This is a joint work with F. De Marchis, A. Malchiodi and P-D. Thizy.

Abstract: I will present a recent result about blow-up asymptotics in the three-dimensional Brezis-Nirenberg problem. For a smooth bounded domain $\Omega \subset \R^3$ and smooth functions $a$ and $V$, we consider a sequence of positive solutions $u_\epsilon$ to $-\Delta u_\epsilon + (a+\epsilon V) u_\epsilon = u_\epsilon^5$ on $\Omega$ with zero Dirichlet boundary conditions, which blows up as $\epsilon \to 0$. We derive the sharp blow-up rate and characterize the location of concentration points in the general case of multiple blow-up. This yields a complete picture of blow-up phenomena in the framework of the Brezis-Peletier conjecture in dimension $N=3$. I will also indicate a forthcoming new result parallel to this one for dimension $N \geq 4$. This is joint work with Paul Laurain (IMJ-PRG Paris and ENS Paris).

Abstract: In this talk we discuss the non-commutative analogue of Neveu decomposition for actions of locally compact amenable groups on finite von Neumann algebras. In addition, we assume $G = \Z_+$ or $G$ is a locally compact group of polynomial growth with a symmetric compact generating set $V$, then for a state preserving action $\alpha$ of $G$ on a finite von Neumann algebra $M$, discuss the convergence in bilateral almost uniformly of the ergodic averages associated with the predual action on $M_{*}$ corresponding to the F\o lner sequence $\{K_n\}_{n \in \N}$ (where $K_n = \{ 0, 1, \ldots n-1 \}$ for $G= \Z_+$ and $K_n = V^n$ otherwise) . At the end, using these results, we establish the stochastic ergodic theorem.

Abstract: In recent years, real convex projective geometric structures (which are a generalization of hyperbolic structures) have played an important role in understanding discrete subgroups of projective general linear groups. This has connections with several other areas like (higher) Teichmüller theory and Anosov representations. In this talk, I will discuss the notion of real convex projective structures and convex co-compact groups and then study them from the perspective of geometric group theory. In particular, I will discuss results (joint work with A. Zimmer) that provide a complete geometric characterization of relatively hyperbolic convex co-compact groups (with respect to any peripheral subgroups).

Abstract: In this talk, I will discuss the question of existence of families of sign-changing solutions to the Yamabe equation, which blow up in the sense that their maximum values tend to infinity. It is known that in the case of positive solutions, there does not exist any blowing-up families of solutions to this problem in dimensions less than 25, except in the case of manifolds conformally equivalent to the round sphere (Khuri, Marques and Schoen, 2009). I will present a construction showing the existence of a non-round metric on spherical space forms of dimensions greater than 10 for which there exist families of sign-changing blowing-up solutions to this problem. Moreover, the solutions we construct have the lowest possible limit energy level. As a counterpart, we will see that such solutions do not exist at this energy level in dimensions less than 10. This is a joint work with Bruno Premoselli (Université Libre de Bruxelles).

Abstract: In collaboration with Yves Colin de Verdière and Luc Hillairet, we study spectral properties of sub-Riemannian Laplacians, which are selfadjoint hypoelliptic operators satisfying the Hörmander condition. Thanks to the knowledge of the small-time asymptotics of heat kernels in a neighborhood of the diagonal, we establish the local and microlocal Weyl law. When the Lie bracket configuration is regular enough (equiregular case), the Weyl law resembles that of the Riemannian case. But in the singular case (e.g., Baouendi-Grushin, Martinet) the Weyl law reveals much more complexity. In turn, we derive quantum ergodicity properties in some sub-Riemannian cases.