In this talk I will explain a construction of equivariant version of Lagrangian Floer homology with compact group action. I will explain some of the ideas to construct it and its (potential) applications.
Tag - Differential geometry
Modelling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to a degenerate elliptic non-linear PDE. A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry.
I will describe joint work-in-progress with Sheel Ganatra and Vivek Shende. We investigate to what extent holomorphic curves and Lagrangian submanifolds can be "localized". One goal is to obtain a combinatorial presentation of the Fukaya category of any Stein manifold. Another corollary of our setup is a local-to-global argument for verifying Abouzaid's generation criterion for the Fukaya category.
The wrapped Fukaya category of an algebraic hypersurface H in (ℂ*)n is conjecturally related via homological mirror symmetry to the derived category of singularities of a toric Calabi-Yau manifold X, whose moment polytope is determined by the tropicalization of H. In this talk we will first explain the statement, and illustrate it for the case of the pair of pants; then we will outline some more speculative ideas about "relative" homological mirror symmetry for pairs ((ℂ*)n, H) and wrapped Fukaya categories of higher-dimensional pairs of pants.
In the 50s John Nash astonished the geometers with his celebrated isometric embedding theorems. A folkloristic explanation of his first theorem is that you should be able to put any piece of paper in your pocket without crumpling or folding it, no matter how large it is. Ten years ago Laszlo, Szekelyhidi, and I discovered unexpected similarities with the behaviour of some classical equations in fluid dynamics. Our remark sparked a series of discoveries and works which have gone in several directions. Among them the most notable is the recent proof of Phil Isett of a long-standing conjecture of Lars Onsager in the theory of turbulent flows.
In this talk I will report on some of the new results in the theory of constant mean curvature (CMC) surfaces M in Riemannian 3-manifolds N. I will mention just a few of the topics touched on in this talk. I first begin with the recent classification of CMC spheres in a homogeneous 3-manifold N and a sketch of its proof. The main result states that any two spheres in N with the same absolute mean curvature differ by an ambient isometry of N. Furthermore, the range of values of the mean curvature spheres are described in terms of the geometry of the universal cover X of N. In the case that X is diffeomorphic to ℝ3 then there exists a sphere of constant mean curvature H in N iff H is greater than half the Cheeger constant of X and otherwise there exists a sphere of constant mean curvature in N for every real number. These results generalize previous work of Hopf, of Abresch-Rosenberg and more recently of Danieil-Mira and of Meeks in the case of the Sol geometry.
Jointly with Tinaglia, we obtain curvature for embedded disks of fixed constant mean curvature H>0 in any fixed homogeneous 3-manifolds. In the ℝ3 setting this result implies that any complete embedded finite topology surface in ℝ3 of constant mean curvature is proper; this generalizes the previous work of Colding -Minicozzi in the case of minimal surfaces. Previous classification results then imply that the only complete embedded simply connected constant mean curvature surfaces in ℝ3 are the plane, the helicoid and round spheres. Another application of this work by Meeks-Tinaglia is to prove that complete embedded CMC surfaces of finite topology in a complete hyperbolic 3-manifold are proper if the mean curvature H is at least 1. On the other hand, Coskunuzer-Meeks-Tinaglia recently constructed for any H in [0,1) a non-proper, complete, stable embedded plane in hyperbolic 3-space having constant mean curvature H.
In 1982 Choi and Wang proved that an embedded closed minimal surface F in the the round 3-sphere S3 has a bound on its area that only depends on its genus; actually their result generalizes from the ambient space S3 to any closed 3-manifold M with positive Ricci curvature. This result was then used by Choi and Schoen to prove the compactness of the moduli space of such examples of fixed genus g in M. Tinaglia and I have been able to give the following related result in the case of connected closed surfaces M embedded in any Riemannian homology 3-sphere manifold N:
Theorem: For any H>0 and non-negative integer g, there exists a constant A(N,g,H) such that any closed surface embedded in M of genus g and constant mean curvature H has area at most A(N,g,H). This area estimate lead to a natural compactification of the moduli space of all such embedded constant mean curvature H examples in N with genus at most g, when H lies in a fixed compact interval [a,b] of positive numbers, and under a compact deformation of the Riemannian metric on N.
The recent classification of properly embedded minimal surfaces of genus 0 in ℝ3 given by Meeks-Perez-Ros, Lopez-Ros, Colin and of Meeks-Rosenberg play a role in the above area estimates, as do the curvature estimates of Meeks-Tinaglia for certain complete embedded CMC surfaces in a Riemannian 3-manifold.
At the end of my talk I will present a brief survey of some recent results on the existence and classification of CMC foliations of closed and non-closed 3-manifolds.
The point of this talk is to present a Lagrangian potential theory, which is in many ways analogous to classical pluripotential theory in complex analysis. I will also introduce a new Lagrangian differential operator of Monge-Ampère type. This ideas are new even in ℂn. However, they apply quite generally, perhaps most importantly to symplectic manifolds equipped with a Gromov metric.
The Lagrangian Monge-Ampère operator is an explicit polynomial on Sym2(TX) whose principle branch defines the space of Lag- harmonics. Interestingly this operator depends only on the Laplacian and the SKEW-Hermitian part of the Hessian if the function. The Dirichlet problem for this operator is solved in both the homogeneous and inhomogeneous cases. It is also solved for each of the other branches. We shall also look at the notions of Lagrangian plurisubharmonic and harmonic functions, Lagrangian convex manifolds and boundaries, and an analogue of the Levi problem. Parallels of this Lagrangian potential theory with standard (complex) pluripotential theory are emphasized.
The starting part for the talk is work of Hitchin, giving a variational description of special geometric structures in dimensions 6, 7, 8 in terms of a volume functional. (See for example "The geometry of 3- forms in six dimensions", JDG 2000.) In the talk we will discuss some developments of this idea, in the case of 7 dimensions and G2 holonomy. In one direction we will consider boundary value problems, and reductions of the G2 equation to 3 and 4 dimensions. In another direction we will consider adiabatic limits, making contact with the theory of maximal submanifolds in spaces of indefinite signature.
Suppose that f is a contractible map from the unit m-sphere to the unit n-sphere with Lipschitz constant L. Is it possible to choose a null-homotopy with Lipschitz constant bounded by a reasonable function of L? Gromov posed this question about twenty years ago. For special choices of m and n, he constructed homotopies with Lipschitz constant at most C(m,n) L. But for most dimensions, the bounds that were known until recently were astronomical, towers of exponentials in L. In the last year, Chambers, Dotterrer, Manin, Ferry, and Weinberger constructed null-homotopies with nearly sharp Lipschitz constants. I will give a little background about the problem and then discuss their work.
Ricci solitons, introduced by R. Hamilton in the mid-1980s, are self-similar solutions to the Ricci flow and often appear as singularity models of the Ricci flow. Ricci solitons are also natural extensions of Einstein metrics and are critical points of certain functionals defined by Perelman and others. In this talk I shall survey some recent developments on gradient shrinking Ricci solitons, including their geometry, classifications, and stability.
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