talks in S98, W98 \ \ \ \ talks in 99-00 \ \ \ \ talks in F00, W01 \ \ \ \ talks in 01-02 \ \ \ \ talks in 02-03 \ \ \ \ talks in 03-04

Differential Geometry Seminar Schedule for Spring 2005

Fridays 3:30 - 4:30pm, SH 6635

4/1 Paolo Cascini, UCSB ``K\"ahler-Ricci Flow and the Minimal Model Program for Projective Varieties"

Abstract: One of the most important problems in Algebraic Geometry is the quest for a Minimal Model for an algebraic variety, that would generalize the classification of algebraic surfaces in higher dimension. In case the variety is of general type, then out of a minimal model one can produce the so called canonical model, a birational model whose canonical bundle is positive. On the other hand, around the 1980's, building on the foundational work of Hamilton in the Riemannian case, H. D. Cao studied the K\"ahler-Ricci flow for canonical metrics on manifolds with definite first Chern class, reproving in particular Calabi's conjecture and the existence of K\"ahler-Einstein metrics in case $c_1 <0$ (the original proof of Calabi's conjecture is due to Yau, who solved it using elliptic methods). We propose to draw a connection between the two theories for projective varieties of general type, and in fact show that, in complex dimension two, the K\"ahler-Ricci flow (starting with a suitable metric) produces the canonical model, generalizing Cao's result.

4/8 Farshid Arjomandi, UCSB ``A Survey of the Rozansky-Witten Invariants"

Abstract: It is a rather well-known fact that Riemannian manifolds whose holonomy group is the unitary group U(n) are Kaehler manifolds and vice versa. Compact irreducible Riemannian manifolds whose holonomy group is the symplectic group Sp(n) are called hyper-Kaehler manifolds. While compact Kaehler manifolds carry only one complex structure, compact hyper-Kaehler manifolds possess three complex structures I, J, and K that behave like the quaternions i, j, and k. Remarkably enough, it turns out that every hyper Kaehler manifold can be turned into a holomorphic symplectic manifold and vice versa. The two main example series of compact hyper Kaehler manifolds include the Hilbert schemes of points on a K3 surface, and also the generalized Kummer varieties.  A new method for studying a hyper-Kaehler manifold X was invented in 1997 by L. Rozansky and E. Witten via associating an invariant to a pair (X,G), where G is a certain kind of graph known as a Jacobi diagram. Shortly after the invention of these invariants, Kapranov and Kontsevich realized that the Riemannian structure is not really necessary for the construction and they showed how the holomorphic structure of X alone is enough for building up these invariants. In fact, the aforementioned quaternionic structures on X not only enable one to use algebraic geometry to study these differential-geometric objects, they also lead to a rich geometry on the algebraic side. More interestingly, it turns out that all the Chern invariants of a holomorphic symplectic manifold are Rozansky-Witten (RW) invariants so that in a sense, RW-theory is a generalisation of the Chern-Weil theory of the characteristic classes.  In this talk I plan to talk about the algebraic approach of Kapranov and Kontsevich, starting out by mentioning some definitions and results on the geometry of compact hyper Kaehler manifolds, followed by a short description of the original 1997 Rozansky-Witten construction. Then I will introduce the Atiyah class, which is the algebraic analog of the Riemann curvature tensor, and the main ingredient in the algebraic build up of the RW-invariants. Next I will define the Jacobi diagrams, their subspaces, ideals, the IHX relation, and the weight systems. Finally I will show how to construct the RW-invariants, using the machinery thus far developed. If time allows, there will also be a quick presentation of an application of the Wheeling Theorem on relating the L2 norm of the Riemann curvature tensor of a compact hyper-Kaehler manifold to some topological quantity, its Todd genus.

4/15 Helena McGahagan, UCSB ``Schr\"odinger Maps"

Abstract: Schr\"odinger maps are solutions of a highly nonlinear PDE with geometric structure arising from the constraint that the solutions must lie on a given manifold. PDE techniques that incorporate the geometry allow us to prove both the existence and uniqueness of Schr\"odinger maps. Geometric problems that arise in the proof of existence include the definition of Sobolev spaces in terms of covariant derivatives and the approximation of maps by smooth maps that lie on the manifold. Also, the use of parallel transport along geodesics to compare solutions yields an improved uniqueness result.

4/22 no meeting

4/29 Xianzhe Dai, UCSB ``Curvature and injectivity radius estimates for Einstein 4-manifolds (apres Cheeger-Tian)"

5/6 Xianzhe Dai, UCSB ``Curvature and injectivity radius estimates for Einstein 4-manifolds (apres Cheeger-Tian), continue"

5/13 no meeting ``"

5/20 Rugang Ye, UCSB ``Curvature and injectivity radius estimates for Einstein 4-manifolds (apres Cheeger-Tian), continue"

5/27 Xianzhe Dai, UCSB ``Curvature and injectivity radius estimates for Einstein 4-manifolds (apres Cheeger-Tian), continue"

6/3 Denis Labutin, UCSB ``Maximal function and weak type (1, 1) inequality"

Differential Geometry Seminar Schedule for Winter 2005

Fridays 3:30 - 4:30pm, SH 6635

1/14 J. Douglas Moore, UCSB ``Towards a Morse Theory for Minimal Surfaces in Riemannian Manifolds"

Abstract: This will be a survey describing some open questions, and recent results that stress the analogies between geodesics and minimal surfaces in Riemannian manifolds. It should be accessible to graduate students who have completed the 240 series.

1/21 Per Tomter, visiting UCSB ``Isometric immersions into complex projective space"

Abstract: There is a somewhat vague conjecture, saying that the principal orbit metrics on a homogeneous space G/H which is a principal orbit, are the only homogeneous metrics that allow an isometric immersion into the Riemannian manifold M on which the compact Lie group G acts. (This has been proved f.ex. for the action of SO(n) =>SO(n)xSO(n) on R(2n) (diagonal embedding); the principal orbits are 2.complex Stiefel manifolds of codimension 3). Here we consider the principal orbits of the isotropy action of a complex projective space CP(n), they are the geodesic 2n-1 spheres. We prove the conjecture for this fundamental case. For n>3, an elaborate use of the Gauss equations will suffice, for n=3 we also need the Codazzi-Mainardi equations, while the most difficult case n=2 reveals that the fundamental theorem of submanifolds does not generalize to CP(n). We need to exploit the fact that the complex structure is parallell in CP(n). We will try to present a survey lecture.

1/28 Will Wylie, UCSB ``Long Homotopies and the Tangent Cone at infinity"

Abstract: In 1968 Milnor conjectured that the fundamental group of a complete manifold with nonnegative Ricci curvature must be finitely generated. While there are many strong partial results supporting the conjecture there is still no complete proof or counterexample. Here we attempt to investigate additional properties of the fundamental group in cases where the fundamental group has already been shown to be finitely generated. In particular, we will show in certain cases that if we are given two homotopic loops in the fundamental group that we can control the length of the homotopy between them based upon the lengths of the curves. We will also discuss the relationship between this geometric control on the fundamental group and special Gromov Hausdorff limits of the manifold called tangent cones at infinity.

2/4 Lei Ni, UCSD ``Ancient solutions to Kahler-Ricci flow"

Abstract: In this talk we discusss the following generalization on a result of Perelman on Ricci flow, that any non-flat ancient solution to K\"ahler-Ricci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero.

2/11 Xianzhe Dai, UCSB ``On the Stability of K\"ahler-Einstein Metrics"

Abstract: In our previous work, we proved stability of a large class of Ricci flat metrics, namely, those with special holonomy. A crucial ingredient is the use of parallel spinor, which dictates that the metric will have to be Ricci flat. In order to deal with general Kahler-Einstein metrics, we found that spin^c is good framework and use it to prove the stability of Kahler-Einstein metrics with nonpositive scalar curvature. As with our previous work, we can use it to draw interesting consequences including a local volume comparison. This is joint work with Xiaodong Wang and Guofang Wei.

2/18 See the special colloquium by Jeff Cheeger ``Degeneration of Metrics with Special Holonomy"

Abstract:

2/25 no meeting

3/4 Doug Moore, UCSB ``Generic Properties of Closed Parametrized Minimal Surfaces in Riemannian Manifolds"

Abstract: A basic technique in the theory of manifolds is transversality theory, based upon Sard's theorem, which allows one to prove, for example, that "almost all" maps from a compact surface into a smooth manifold of dimension at least five are imbeddings. One would like to study some nonlinear partial differential equations by means of calculus on infinite-dimiensional manifolds.  Smale developed an extension of Sard's theorem which has proven to be useful in a wide variety of contexts. In this lecture, we will explore the applications of Smale's extension of Sard's theorem to the theory of minimal surfaces in compact Riemannian manifolds.

3/11 Pengzi Miao, UCSB ``On Boundary Effects of Static Asymptotically Flat $3$-manifolds"

Abstract: An asymptotically flat $3$-manifold with boundary is called static if it arises as a time-symmetric hypersurface in a static spacetime solution to the Vacuum Einstein Equation. The vacuumness and the time-independence of such a spacetime suggest there is no contribution to the ADM mass from either matter distribution or gravitational radiation. Thus, it has been expected that the only mass contribution comes from the boundary. In this talk, we give some uniqueness theorem of static asymptotically flat manifolds with boundary, established following the above physical spirit. We also talk about their relation to the Riemannian Penrose Inequality and Bartnik's quasi-local mass definition.

Differential Geometry Seminar Schedule for Fall 2004

Fridays 3:30 - 4:30pm, SH 6635

9/24 organizational meeting

10/1 Pengzi Miao, UCSB ``On Small Data Solution to Static Metric Extension Problem"

Abstract: Motivated by R. Bartnik's quasi-local mass definition in general relativity, we study the small data solution to the static metric extension conjecture. We first recall the basic definition and property of static metrics. Then we state the static metric extension problem with emphasis on explaining both the interior equation and the boundary condition. In the end we present an existence result on small data solution. Open questions on relation among static metrics, Einstein metrics and constant scalar curvature metrics may also be discussed.

10/8 Pengzi Miao, UCSB ``On Static Metric Extension in General Relativity II"

Abstract: We introduced the static metric extension problem proposed by R. Bartnik in the previous talk. In this talk we will give it rigorous mathematical analysis. First, we will analyze static metrics from a scalar curvature deformation point of view and state J. Corvino's theorem on local scalar curvature deformation. Then we will reformulate our variational problem to prove that a mass minimizer, if exists, must statisfy the system of static equations and the expected boundary condition. Then we will focus on small data solutions for compact domains in R^3 and prove that there always exists solution to the extension problem if the metric deformation of standard balls satisfies a symmetry condition.

10/15 Xiaofeng Sun, Harvard Univ. ``Canonical metrics on the moduli space of Riemann surfaces"

Abstract: We study two new complete Kahler metrics, the Ricci metric and perturbed Ricci metric on the moduli space of closed Riemann surfaces and show that these metrics have nice curvature properties. Both of these metrics have bounded geometry and have Poincare growth near the boundary of the moduli space. Furthermore, the holomorphic sectional curvature and the Ricci curvature of the perturbed Ricci metric are pinched between negative constants. By using these new metrics we showed that all known complete metrics on the Teichmuller space and moduli space are equivalent in the sense that they are quasi-isometric. This proved the conjecture of Yau about the equivalence between the Teichmuller metric and the Cheng-Yau-Mok Kahler-Einstein metric and the conjecture of Bers about the equivalence between the Kobayashi metric and the Bergman metric. We then investigate the behavior of the Kahler class of the Kahler-Einstein metric and derive algebro-geometric properties of the moduli space. As a direct consequence, we showed that the moduli space is of log general type. More importantly, the logarithm contangent bundle of the moduli space is Mumford stable. Also, following Yau's estimates, we show that the Kahler-Einstein metric has bounded geometry. This is a joint work with Prof. Kefeng Liu and Prof. Yau.

10/22 (cancelled) Doug Moore, UCSB ``Bumpy metrics for minimal surfaces"

Abstract: Morse theory on finite-dimensional manifolds establishes inequalities between the number of critical points of generic function on a compact manifold M and the Betti numbers of M. In the Morse theory of closed geodesics in a compact manifold M, one cannot modify the action function, so instead one chooses a generic metric on M. Abraham proved that for a generic choice of metric, nonconstant closed geodesics lie on nondegenerate critical submanifolds of dimension one. Using this theorem one can establish Morse inequalities between the number of critical submanifolds and Betti numbers of the free loop space of M. This lecture explores the properties of parametrized minimal surfaces in compact manifolds with generic metrics. It turns out that for generic metrics, prime minimal two-spheres lie on nondegenerate critical submanifolds of dimension six, prime minimal tori lie on such submanifolds of dimension two, while minimal surfaces of higher genus are nondegenerate in the usual sense. Some simple applications of this result will be described.

10/29 Xiaoling Wang, UCSB ``Modular Invariance and Witten's Anomaly Cancellation Formula"

Abstract: To research super string theory, in 1983, the physicists Alvarez-Gaume and Witten found a so-called "miraculous cancellation" which reveals a beautiful relation between the Hirzebruch L-form and the Hirzebruch-¬-form of a 12-dimensional Riemannian manifold. Kerfeng Liu established higher dimensional "miraculous cancellation" fornulas for (8k+4)-dimensional smooth manifolds by developing modular invariance properties of characteristic forms. For each (8k+2)-dimensional closed Riemannian manifolds with a rank two real oriented vector bundle over it, Weiping Zhang and Fei Han established a cancellation formulas on the level of Characteristic classes. In fact, it actually holds on the level of differential forms, and I will give a direct proof by using the modular invariance method developed by Liu. We then will obtain more cancellation formulas for even dimensional manifolds with a complex line bundle involved.

11/5 Jingyi Chen, UBC, visiting Stanford University ``Mean curvature flow of calibrated submanifolds"

11/12, Pengzi Miao, UCSB ``Remark on static, asymptotically flat and scalar flat 3-manifolds with horizon boundary"

Abstract: Given a static asymptotically flat and scalar flat 3-manifold M with non-empty boundary, a classic result of Bunting and Masood-ul Alam states that if the static potential function vanishes on the boundary, then M is isometric to the half Schwarzschild manifold. Motivated by Bartnik's minimal mass extenstion conjecture, we discuss how the boundary assumption on the static potential function can be replaced by geometric boundary conditions of the manifold.

11/19 no meeting

11/26 Thanksgiving

12/3 no meeting

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