Differential Geometry Seminar

In Fall 2013, we will meet on Tuesdays, at 2:45pm in Room 3207. The organizers of this seminar are Dan Lee, Zheng Huang, and Marcello Lucia. Please email Dan at dan.lee(NoSpamPlease)qc.cuny.edu to schedule a guest speaker. The CUNY Graduate Center is located at 365 Fifth Avenue at 34th Street, diagonally across the street from the Empire State Building, just two blocks from Penn Station (NYC).

Fall 2013:

• Tuesday, 9/24 at 1:30 (Math Thesis Room): Goncalo Oliveira (Imperial College)
  Monopoles in Higher Dimensions
  Abstract:: The Monopole (Bogomolnyi) equations are Geometric PDE in 3 dimensions that admit generalizations to Higher dimensional manifolds with special structures on them. Calabi Yau and G_2 manifolds are the main candidates for interesting solutions to these equations. There are several conjectural relations between the submanifold theory of such manifolds and solutions to these PDE. These include theories counting calibrated cycles and defining invariants of Calabi Yau and G_2 structures. However, up to date there are no known interesting solutions of the monopole equations. In this talk, I plan to describe these monopole equations from scratch and also announce a construction of solutions to these.
  
  
• Tuesday, 9/24 at 2:45 (2nd talk of double header, Room 3207): Anda Degeratu (University of Freiburg)
  Witten spinors on nonspin manifolds
  Abstract: Unlike a 3-dimensional manifold, a higher dimensional manifold need not be spin. On an oriented Riemannian manifold the obstruction to having a spin structure is given by the second Stiefel-Whitney class. I will show that even when this obstruction does not vanish, it is still possible to define a notion of singular spin structure and associated singular Dirac operator. Then, modeling on Witten's proof of the Positive Mass Theorem, I will define the notion of Witten spinor on an asymptotically flat nonspin manifold, show their existence and describe their properties.
  
  
• Tuesday, 10/1: Lu Wang (Johns Hopkins University)
  Uniqueness of asymptotically conical shrinking Ricci solitons
  Abstract: Shrinking Ricci solitons generalize positive Einstein manifolds and play a central role in the analysis of singularities of the Ricci flow. At present, essentially all known complete noncompact examples are either locally reducible as products or possess conical structures at infinity. I will describe recent joint work with Brett Kotschwar in which we investigate the rigidity of such conical structures and show that, if two shrinking solitons are asymptotic to the same cone along some end of each, then the solitons must actually be isometric on some neighborhoods of infinity of these ends. As an application, we prove that the only complete shrinker asymptotic to a rotationally symmetric cone is the Gaussian soliton.
  
  
• Tuesday, 10/8: Ran Ji (CUNY Graduate Center)
  The asymptotic Dirichlet problems on manifolds with negative curvature
  Elton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable under the curvature conditions $-C e^{2-\eta}r(x) \leq K_M(x)\leq -1$ with $\eta>0$. We give an analytical proof of the same statement by a modification of an argument due to M. T. Anderson and R. Schoen.
  
  
• Tuesday, 10/15: no seminar
  Columbus Day
  
  
• Tuesday, 10/22: Paolo Piccione (University of Sao Paulo)
  Delaunay type hypersurfaces in cohomogeneity one manifolds Abstract: Classic Delaunay hypersurfaces are constant mean curvature (CMC) rotationally symmetric surfaces in Euclidean space, which were classified by Delaunay in 1841. They can be spheres, unduloids, catenoids, nodoids or cylinders. Similar constructions of rotationally symmetric CMC hypersurfaces exist in other space forms, in particular, in round spheres. In this talk, I will explain how a bifurcation approach can be used to construct Delaunay type hypersurfaces in a more general class of ambients, namely, cohomogeneity one manifolds. This construction yields infinitely many geometrically distinct CMC hypersurfaces with same topology as the principal orbit, and can be geometrically understood in terms of a partial break of symmetry. In particular, we recover the known construction on spheres (matching existence classification results in low dimensions), and provide a large class of new examples. This is joint work with Renato G. Bettiol (Notre Dame).
  
  
  
• Tuesday, 10/29: Brian Benson (University of Illinois at Urbana-Champaign)
  Sturm-Liouville Estimates for the Spectrum and Cheeger Constant
  Abstract: Buser's inequality gives an upper bound on the first non-zero eigenvalue of the Laplacian of a closed manifold $M$ in terms of the Cheeger constant $h(M)$. Agol later gave a quantitative improvement of Buser's inequality. Agol's result is less transparent since it is given implicitly by a set of equations, one of which is a differential equation Agol could not solve except when $M$ is three-dimensional. We show that a substitution transforms Agol's differential equation into the Riemann differential equation. Then, we give a proof of Agol's result and also generalize it using Sturm-Liouville theory. Under the same assumptions on $M$, we are able to give upper bounds on the higher eigenvalues of $M$, $\lambda_k(M)$, in terms of the eigenvalues of Sturm-Liouville problems which depend on $h(M)$. We then compare the Weyl asymptotic of $\lambda_k(M)$ given by the works of Cheng, Gromov, and B\'erard-Besson-Gallot to the asymptotics of our Sturm-Liouville problems given by Atkinson and Mingarelli.
  
  
• Tuesday, 11/5: Ali Aleyasin (Stony Brook University)
  The space of asymptotically locally euclidean Kähler potentials
  Abstract: Let $H$ be the space of Kähler metrics in a fixed cohomology class. This space may be endowed with a Weil-Petersson-type metric, referred to as the Mabuchi metric, which allows one to study the geometry of $H$. After the works of Mabuchi and Donaldson, it is now well-known that the geometry of the space of Kähler potentials, in particular, the geodesics in $H$, may be used for studying `canonical metrics' on the base manifold. The geodesics are indeed solutions to a homogeneous complex Monge-Ampère equation. In order to be interpreted as the potential of a Kähler metric however, one needs to prove some regularity for such solutions. X-X. Chen first proved such regularity results for compact manifolds. Using different approach, I shall prove weighted estimates for the spatial and temporal derivatives of solutions in the case of ALE Kähler potentials. As application, we prove the boundedness from below of the Mabuchi energy. Further, we give an alternative proof of the uniqueness of ALE Kähler metrics of constant scalar curvature.
  
  
• Tuesday, 11/12: Hung Tran (Cornell)
  The Weyl Tensor of Gradient Ricci Solitons
  Abstract: We derives several new identities for the Weyl tensor of gradient Ricci solitons, specifically in dimension four. In particular, in the first part, we prove a Bochner-Weitzenb\"ock type formula for the norm of the self-dual Weyl tensor which has several applications including a gap theorem. In the second part, we are mostly concerned with the interaction of different components of curvature, the gradient and Hessian of the potential function. The Weyl tensor arises naturally in these investigations as it is the main distinction in higher dimensions. Applications here are rigidity results.
  
  
• Tuesday, 11/19: no seminar
  
  
• Tuesday, 11/26: no seminar
  
  
• Tuesday, 12/3: Robert Haslhofer (New York University)
  Mean curvature flow of mean convex hypersurfaces
  Abstract: In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow. In the present talk, we explain a new treatment of the theory of mean convex (and k-convex) flows. This includes: (1) an estimate for derivatives of curvatures, (2) a convexity estimate, (3) a cylindrical estimate, (4) a global convergence theorem, (5) a structure theorem for ancient solutions, and (6) a partial regularity theorem. Our new proofs are both more elementary and substantially shorter than the original arguments. Our estimates are local and universal. A key ingredient in our new approach is the new non-collapsing result of Andrews. Some parts are also inspired by the work of Perelman. This is joint work with Bruce Kleiner.
  

Past Seminar Schedules and Abstracts:

Spring 2013 , Fall 2012 , Spring 2012 , Fall 2011 , Spring 2011 , Fall 2010 , Spring 2010, Fall 2009, Spring-Summer 2009, Fall 2008, Spring 2008, Fall 2007, Spring-Summer 2007, Fall 2006, Spring-Summer 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001.