Differential Geometry Seminar at UCSB
Fridays 3:00 - 3:50pm, SH 6635
1/23 Changyong Yin (UCLA) “Quantum Correction and the Moduli Spaces of Calabi-Yau Manifolds”
Abstract: We introduce an new notation— quantum correction for the Teichmuller space of polarized Calabi-Yau manifolds. Under the assumption of vanishing of weak quantum correction, we show that the Teichmuller spaces, with the Weil-Petersson metric, are locally Hermitian symmetric spaces. For Calabi-Yau threefolds, we prove that vanishing of strong quantum correction is equivalent to that the image of the Teichmuller space under the period map is an open submanifold of a globally Hermitian symmetric space.
Abstract: We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature — including strictly convex domains of the Euclidean space R^3.
2/13 Zhizhang Xie (Texas A&M University) "Higher signature on Witt spaces"
Abstract: The signature is a fundamental homotopy invariant for topological manifolds. However, for spaces with singularities, this usual notion of signature ceases to exist, since, in general, spaces with singularities fail the usual Poincaré duality. A generalized Poincaré duality theorem for spaces with singularities was proven by Goresky and MacPherson using intersection homology. The classical signature was then extended to Witt spaces by Siegel using this generalized Poincaré duality. Witt spaces are a natural class of spaces with singularities. For example, all complex algebraic varieties are Witt spaces. In this talk, I will describe a combinatorial approach to the higher signature of Witt spaces, using methods of noncommutative geometry. This is based on joint work with Nigel Higson.
2/20 Ben Schmidt (Michigan State University) "Three-manifolds with many flat planes."
Abstract: The Euclidean rank of a complete geodesic c(t) in a Riemannian manifold M is the dimension of the space of parallel Jacobi fields along c(t). The Euclidean rank of M is the least rank of its geodesics. Note that each geodesic c(t) has rank at least one since the velocity field c'(t) is parallel.
I'll motivate and describe the proof of the following rigidity theorem: A complete Riemannian three-manifold has Euclidean rank at least 2 if and only if its universal covering is isometric to a Riemannian product.
Based on joint work with Renato Bettiol.
3/6 Gye Seon Lee (Ruprecht-Karls-Universität Heidelberg) "Andreev's theorem on projective Coxeter polyhedra"
Abstract: In 1970, E.M. Andreev gave a full description of 3-dimensional compact hyperbolic polyhedra with dihedral angles submultiples of pi. We call them hyperbolic Coxeter polyhedra. More precisely, given a combinatorial polyhedron C with assigned dihedral angles, Andreev’s theorem provides necessary and sufficient conditions for the existence of a hyperbolic Coxeter polyhedron realizing C. Since hyperbolic geometry arises naturally as sub-geometry of real projective geometry, we can ask an analogous question for compact real projective Coxeter polyhedra. In this talk, I’ll give a partial answer to this question. This is a joint work with Suhyoung Choi.
3/10 Manuel Amann (Karlsruhe Institute of Technology) "Complex topology"
Abstract: [please note the nonstandard time & place] It is an interesting question to determine how complex it is to actually compute topological invariants of certain spaces from a suitable algebraic model. For example, it is known that computing the Betti numbers of topological spaces can be NP-hard. In this talk I shall approach the following concrete problems of that kind: Given a simply-connected space X with both H∗(X; Q) and π∗(X)⊗Q being finite-dimensional, what is the computational complexity of an algorithm computing the cup-length and the rational Lusternik–Schnirelmann category of X? How complicated is it to decide whether a Lie group action on a compact manifold has only finite isotropy, i.e. is almost free? Basically, by a reduction from the decision problem whether a given graph is k-colourable for k ≥ 3 it will be shown that even stricter versions of these problems are NP-hard.
3/13 Qi Zhang (UC, Riverside) "Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature"
Abstract: We consider Ricci flows that satisfy certain scalar curvature bounds. It is found that the time derivative for the solution of the heat equation and the curvature tensor have better than expected bounds. Based on these, we derive a number results. They include: bounds on distance distortion at different times and Gaussian bounds for the heat kernel, backward pseudolocality, $L^2$-curvature bounds in dimension $4$. This is a joint work with Richard Bamler.
Abstract: The eigenvalues of the Laplacian encode fundamental geometric information about a Riemannian metric. As an example of their importance, I will discuss how they arose in work of Cao, Hamilton and Illmanan, together with joint work with Stuart Hall, concerning stability of Einstein manifolds and Ricci solitons. I will outline progress on these problems for Einstein metrics with large symmetry groups. We calculate bounds on the first non-zero eigenvalue for certain Hermitian-Einstein four manifolds. Similar ideas allow us estimate to the spectral gap (the distance between the first and second non-zero eigenvalues) for any toric Kaehler-Einstein manifold M in terms of the polytope associated to M. I will finish by discussing a numerical proof of the instability of the Chen-LeBrun-Weber metric.
5/1 Qing Han (Notre Dame) "Refined boundary expansions of minimal surfaces in the hyperbolic space"
Abstract: The minimal surface equation in the hyperbolic space is given by a quasilinear elliptic equation, which is non-uniformly elliptic and becomes singular on the boundary. In this talk, we discuss a recent result of the optimal boundary expansions of minimal surfaces in the context of the finite regularity.
5/8 Double seminar -- special times: 3pm and 4pm:
Longzhi Lin (UC, Santa Cruz) "Star-shaped mean curvature flow"
Abstract: A one-parameter family of hypersurfaces in Euclidean space evolves by mean curvature flow if the velocity at each point is given by the mean curvature vector. It can be viewed as a geometric heat equation, i.e., it is locally moving in the direction of steepest descent for the volume element, deforming surfaces towards optimal ones (minimal surfaces). In this talk we will discuss some recent work on the local curvature estimate and convexity estimate for the star-shaped mean curvature flow and the consequences. In particular, star-shaped MCF is generic in the sense of Colding-Minicozzi. This is joint work with Robert Haslhofer.
Zhenlei Zhang (Capital Normal University -- visiting Princeton University) "On the convergence of Kähler-Ricci flow on smooth minimal models of general type"
Abstract: In this talk, i will present the Cheeger-Gromov convergence of Kahler-Ricci flow on smooth minimal models of general type when the dimension is less or equal to 3. The result relies on a uniform integral estimate of the Ricci curvature under the Kahler-Ricci flow. It is a joint work with Professor Tian.
5/15 Anna Siffert (University of Pennsylvania) "Harmonic maps of cohomogeneity one manifolds"
Abstract: In my talk I present a method for reducing the problem of constructing harmonic self-map of cohomogeneity one manifolds to solving non-standard singular boundary value problems for non-linear ordinary differential equations. Furthermore, I introduce new techniques to finding solutions of these boundary value problems and discuss several interesting examples.
5/22 Sho Seto (University of California, Irvine) "On the asymptotic expansion of the Bergman kernel"
Schedule for Fall 2014
Fridays 3:00 - 3:50pm, SH 6635
9/8, 2pm, Valentino Tosatti (Northwestern University) Collapsing of Calabi-Yau manifolds
Abstract:I will discuss the problem of understanding how Ricci-flat Calabi-Yau manifolds collapse to lower-dimensional spaces, and how this is relevant to the Strominger-Yau-Zaslow picture of mirror symmetry. I will present some results in this direction, which are joint work with M. Gross, Y. Zhang, with H.J. Hein and with B. Weinkove, X. Yang.
10/10, Gang Liu (UC, Berkeley) Three circle theorems on Kahler manifolds and applications
Abstract: The classical Hadamard Three Circle theorem is generalized to complete Kahler manifolds with nonnegative holomorphic sectional curvature. Various applications will be discussed. For example, the connection with Yau's uniformization conjectures; the resolution of Ni's conjecture on complete Kahler manifolds with nonnegative bisectional curvature.
10/17, William Wylie (Syracuse University) Gradient shrinking Ricci solitons of half harmonic Weyl curvature
Abstract: Ricci solitons are self similar solutions to the Ricci flow and a natural generalization of Einstein metrics. In this talk I'll survey some prior results about the classification of gradient shrinking Ricci solitons under varying conditions on dimension, curvature, or symmetry. Then I'll discuss a new classification result of 4-dimensional gradient shrinking Ricci solitons with half harmonic Weyl curvature. This is joint work with J.Y. Wu of Shanghai Maritime University and P. Wu of Cornell.
11/7, Lee Kennard (UCSB) Positive sectional curvature for manifolds with density
Abstract: Any positive function on a manifold is called a density, as it can be interpreted as a mass density or probability distribution. If the manifold is Riemannian, notions of scalar curvature (introduced by Perelman) and Ricci curvature (introduced by Lichnerowicz) for manifolds with density have been intensely studied. Recently William Wylie (Syracuse University) refined these notions by defining sectional curvature for manifolds with density and proving that a large number of classical results involving sectional curvature carry over to the case with density. In recent work, Wylie and I investigate the concept of positive sectional curvature in this context. I will define these notions, discuss elementary examples, and survey some of our recent results.
11/14, Changliang Wang (UCSB) An eigenvalue problem for compact manifolds with isolated conical singularities
Abstract: I will talk about some work with Prof. Xianzhe Dai about trying to generalize Perelman's lambda-functional on manifolds with isolated conical singularities. This is essentially an eigenvalue problem and closely related to the asymptotic behavior of eigenfunctions near singularities.
12/5, Andrew Cotton-Clay (UCSB) Holomorphic Curves in Mapping Tori
Abstract: We discuss counts of pseudo-holomorphic curves in R times a mapping torus and applications to fixed points and geometry in 2-and 3-dimensions.