Related papers: Critical Metrics for Riemannian Curvature Function…
This is an expository article based on my lectures on eigenfunctions of the Laplacian for the 2013 IAS/Park City Mathematics Institute (PCMI) summer school in geometric analysis. Many of the results are based on joint work with H.…
These are the notes corresponding to the course given at the IAS-Park City graduate summer school in July 2007.
We study closed $n$-dimensional manifolds of which the metrics are critical for quadratic curvature functionals involving the Ricci curvature, the scalar curvature and the Riemannian curvature tensor on the space of Riemannian metrics with…
In this paper, we investigate a class of quadratic Riemannian curvature functionals on closed smooth manifold $M$ of dimension $n\ge 3$ on the space of Riemannian metrics on $M$ with unit volume. We study the stability of these functionals…
In this paper we prove rigidity results on critical metrics for quadratic curvature functionals, involving the Ricci and the scalar curvature, on the space of Riemannian metrics with unit volume. It is well-known that Einstein metrics are…
This chapter is based on lectures on Randomized Numerical Linear Algebra from the 2016 Park City Mathematics Institute summer school on The Mathematics of Data.
Construction of immersions with "small" curvatures between Riemannian manifolds and indicating obstructions to such immersions
An article based on a four-lecture introductory minicourse on minimal surface theory given at the 2013 summer program of the Institute for Advanced Study and the Park City Mathematics Institute.
We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some…
We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold $M$ given by $\mathcal{R}_{\frac{n}{2}}(g):= \int_M |R(g)|^{\frac{n}{2}}dv_g$ where $R(g)$, $dv_g$ denote the…
These are notes related to a 12-hour course of lectures given at the Centre de Recerca Mathem\`atica near Barcelona in February, 2010. The aim of the course was to explain results on curves and their Jacobians over function fields, with…
This paper agrees basically with the talk of the author at the workshop "Homological Mirror Symmetry and Applications", Institute for Advanced Study, Princeton, March 2007.
This note surveys some classical results and recent developments on the interplay between lower curvature bounds and the isoperimetric problem. It is based on mini-courses given at the European Doctorate School of Differential Geometry…
These are lecture notes for lectures at the Park City Math Institute, summer 2007. We cover aspects of the dimer model on planar, periodic bipartite graphs, including local statistics, limit shapes and fluctuations.
We discuss some properties of the distance functions on Riemannian manifolds and we relate their behavior to the geometry of the manifolds. This leads to alternative proofs of some "classical" theorems connecting curvature and topology.
Some examples of three-dimensional metrics of constant curvature defined by solutions of nonlinear integrable differential equations and their generalizations are constructed. The properties of Riemann extensions of the metrics of constant…
In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}. Quasi-$k$-curved metrics generalize the metrics of space forms. We construct explicit…
We overview main topics and ideas in spaces with their scalar curvatures bounded from below, and present a more detailed exposition of several known and some new geometric constraints on Riemannian spaces implied by the lower bounds on…
This paper is the third in a series dedicated to the fundamentals of sub-Riemannian geometry and its implications in Lie groups theory: "Sub-Riemannian geometry and Lie groups. Part I", math.MG/0210189, available at…
This article was prepared in connection with the 2009 Barnett lecture at the University of Cincinnati, and deals with various classes of fractal sets and analysis on them.