Geometry and Analysis on Manifolds 2017
University of California, Santa Barbara
April 7-9, 2017

• Richard Bamler: Uniqueness of weak solutions to the Ricci flow

  Abstract: In his resolution of the Poincaré and Geometrization Conjectures, Perelman constructed Ricci flows in which singularities are removed by a surgery process. His construction depended on various auxiliary parameters, such as the scale at which surgeries are performed. At the same time, Perelman conjectured that there must be a canonical flow that automatically "flows through its surgeries?, at an infinitesimal scale. Recently, Kleiner and Lott constructed so-called Ricci flow space-times, which exhibit this desired behavior. In this talk, I will first review their construction. I will then present recent work of Bruce Kleiner and myself, in which we show that these Ricci flow space-times are in fact unique and fully determined by their initial data. Therefore, these flows can be viewed as ?canonical?, hence confirming Perelman?s Conjecture. I will also discuss further applications of this uniqueness statement.

• Jeffrey Case : The Q-prime curvature in CR geometry, and related CR invariant operators

  Abstract: There are many ways in which the Q-prime curvature is the natural CR analogue of the Q-curvature from conformal geometry. For example, it arises in the sharp Moser--Trudinger inequality and its total integral is a CR invariant which is closely related to the Gauss--Bonnet formula; in particular, it has topological applications. These connections are well-understood in dimension three, with many questions remaining in higher dimensions. I will describe a new construction of the Q-prime curvature and its potential applications to such questions in higher dimensions. This is joint work with Rod Gover.

• Sebastian Goette: Singular $\eta$-forms and odd $K$-theory

  Abstract: Let $p\colon M\to B$ be a proper submersion with odd-dimensional fibers, and let $(D_b)_{b\in B}$ be a smooth family of fiberwise Dirac operators. Assume that $0\ne\mathrm{ind}(D)\in K^1(B)$, then $\ker(D)$ cannot form a vector bundle over $B$. In her phd-thesis, Anja Wittmann investigated the case where $\dim(D_b)\le 1$ for all $b\in B$, and where $0$ is a regular value of the family of eigenvalues of~$D_b$. In this case, the associated $\eta$-form becomes a current on $B$ whose exterior derivative gives rise to a family index theorem.

• Xiaonan Ma: Generalized Bergman kernels on symplectic manifolds and applications

  Abstract: A suitable notion of ``holomorphic section'' of a prequantum line bundle on a compact symplectic manifold is the eigensections of low energy of the Bochner Laplacian acting on high $p$-tensor powers of the prequantum line bundle. We explain the asymptotic expansion of the corresponding kernel of the orthogonal projection as the power p tends to infinity. This implies the compact symplectic manifold can be embedded in the corresponding projective space. With extra effort, we show the Fubini-Study metrics induced by these embeddings converge at speed rate $1/p^{2}$ to the symplectic form. We explain also its implication on Bezerin-Toeplitz quantizations.

• Lei Ni: Isoperimetric inequalities via the viscosity

• Jie Qing: Strong rigidity for asymptotically hyperbolic Einstein manifolds

  Abstract: In this talk I will talk on our recent work on asymptotically hyperbolic Einstein manifolds. I will present a proof for a sharp volume comparison theorem for asymptotically hyperbolic Einstein manifolds, which will imply not only the rigidity theorem for hyperbolic space in general dimension but also curvature estimates for asymptotically hyperbolic Einstein manifolds. In particular, as a consequence of our curvature estimates, one now knows that the asymptotically hyperbolic Einstein metrics with conformal infinities of sufficiently large Yamabe constant have to be negatively curved.

• Julie Rowlett: A Polyakov formula for angular variations

  Abstract: Polyakov's formula expresses a difference of zeta-regularized determinants of Laplace operators, an anomaly of global quantities, in terms of simple local quantities. Such a formula is well known in the case of closed surfaces (Osgood, Philips, & Sarnak 1988) and surfaces with smooth boundary (Alvarez 1983). Due to the abstract nature of the definition of the zeta-regularized determinant of the Laplacian, it is typically impossible to compute an explicit formula. Nonetheless, Kokotov (genus one Kokotov & Klochko 2007, arbitrary genus Kokotov 2013) demonstrated such a formula for polyhedral surfaces! I will discuss joint work with Clara Aldana concerning the zeta regularized determinant of the Laplacian on Euclidean domains with corners. We determine a Polyakov formula which expresses the dependence of the determinant on the opening angle at a corner. Our ultimate goal is to determine an explicit formula, in the spirit of Kokotov's results, for the determinant on polygonal domains, and the results which shall be presented here are the crucial first steps towards such a formula.

• Sho Seto: First eigenvalue of the Laplacian on closed manifolds

  Abstract: In this talk we will give a survey of results and recent progress on estimates for the first Neumann eigenvalue of the p-Laplacian of closed manifolds with integral curvature conditions (joint work with Guofang Wei) and on the fundamental Dirichlet gap for the Laplacian for convex domains in constant curvature manifolds (joint work with Lili Wang and Guofang Wei).

• Changliang Wang: Stability of Riemannian manifolds with Killing spinors

  Abstract: Einstein metrics on a compact manifold are critical points of the normalized total scalar curvature functional. So it is natural to study the behavior of the second variation of the normalized total scalar curvature functional at an Einstein metric. This is known as the linear stability problem of Einstein metrics. Riemannian manifolds with non-zero Killing spinors are Einstein. We prove that complete manifolds with non-zero imaginary Killing spinors are stable by using a Bochner type formula, which was proved by McKenzie Wang and then was rediscovered by Xianzhe Dai, Xiaodong Wang, and Guofang Wei. This stability result has already been proved by Klaus Kroncke in a different way. Moreover, existence of real Killing spinors is closely related to the Sasaki-Einstein structure. A regular Sasaki-Einstein manifold is essentially the total space of a certain principal circle bundle over a Kahler-Einstein manifold. We prove that if the base space is a product of two Kahler-Einstein manifolds then the regular Sasaki-Einstein manifold is unstable. This provides us many new examples of unstable manifolds with real Killing spinors. More generally, we prove that Einstein metrics on principal torus bundles constructed by McKenzie Wang and Wolfgang Ziller are unstable if the base is a product of Kahler manifolds.

• Yi Wang: A fully nonlinear Sobolev trace inequality

  Abstract: The $k$-Hessian operator $\sigma_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $\sigma_k(D^2 u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $\int -u \sigma_k(D^2 u)$. We construct a natural boundary functional which, when added to the $k$-Hessian energy, yields as its critical points solutions of $k$-Hessian equations with general non-vanishing boundary data. As a consequence, we prove a sharp Sobolev trace inequality for $k$-admissible functions $u$ which estimates the $k$-Hessian energy in terms of the boundary values of $u$. This is joint work with Jeffrey Case.

• Peng Wu: On the first eigenvalue of Einstein four-manifolds with nonnegative curvature

  Abstract: A classical result of Lichnerowicz states that for an n-dimensional manifold with $Ric\geq g$, the first eigenvalue of the Laplace operator is bounded below by $\frac{n}{n-1}$. In this talk we will show that for the case of Einstein four-manifolds with nonnegative sectional curvature, the first eigenvalue is bounded above by $\frac{4}{3}+\sqrt[3]{4}$.

• Will Wylie: Sectional Curvature for manifolds with density

  Abstract: The theory of Ricci curvature for manifolds with density (or measure) has been well studied. Until recently, there has not been a corresponding theory of weighted sectional curvatures. We aim to fill this gap by approaching the geometry of manifolds with density by studying a natural torsion free affine connection. We obtain generalizations of many of the classical comparison results for sectional curvature to manifolds with density such as the (non-smooth) 1/4-pinched sphere theorem, Cheeger type finiteness theorems, the theorems of Cartan-Hadamard and Preissman for non-positive curvature, and classification results for positively curved spaces with symmetry of Grove-Searle and Wilking. This is joint work with Lee Kennard (Univ. of Oklahoma) and Dmytro Yeroshkin (Idaho State).

• Xin Zhou: Min-max minimal hypersurfaces with free boundary

  Abstract: I will present a joint work with Martin Li. Minimal surfaces with free boundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free boundary were constructed using the min-max method. Our result allows the min-max free boundary minimal hypersurface to be improper; nonetheless the hypersurface is still regular.

Hotels:

We recommend the following hotels. Please mention the Conference and the UCSB Department of Mathematics at the time of making the reservation.

The hotels have shuttles to campus.