Let Z be a compact, connected 3-dimensional complex manifold with vanishing first and second Betti numbers and non-vanishing Euler characteristic. We prove that there is no holomorphic mapping from Z onto any 2-dimensional complex space. Combining this with a result of Campana-Demailly-Peternell, a corollary is that any holomorphic mapping from the 6-dimensional sphere S^6, endowed with any hypothetical complex structure, to a strictly lower-dimensional complex space must be constant. In other words, there does not exist any holomorphic fibration on S^6. This is joint work with Nobuhiro Honda.

Talk Schedule (each talk 30-35 minutes)

2:00 PM   Alex Mramor (University of Copenhagen)

2:45 PM   Hongyi Sheng (UC San Diego)

3:30 PM   Kai-Wei Zhao (Notre Dame)

4:15 PM   Tin Yau Tsang (New York University)

5:00 PM   Xiaolong Li (Wichita State University)

Titles/Abstracts

Speaker: Alex Mramor (University of Copenhagen)
Title: On the Unknottedness of Self Shrinkers
Abstract: The mean curvature flow, the natural analogue of the heat equation in submanifold geometry, often develops singularities and roughly speaking these singularities are modeled on self shrinkers, which are surfaces that give rise to mean curvature flows that move by dilations. it happens that self shrinkers are minimal surfaces in a metric which, while poorly behaved, is Ricci positive in a certain sense so it is natural, for instance, to ask what type of qualities shrinkers have in common with minimal surfaces in the round 3-sphere. Inspired by an old work of Lawson on such surfaces in this talk we discuss some unknottedness results for self shrinkers in R^3, some of which are joint work with S. Wang.

Speaker: Hongyi Sheng (UC San Diego)
Title: TBA
Abstract: TBA

Speaker: Kai-Wei Zhao (Notre Dame)
Title: Uniqueness of Tangent Flows at Infinity for Finite-Entropy Shortening Curves
Abstract: Curve shortening flow is, in compact case, the gradient flow of arc-length functional. It is the simplest geometric flow and is a special case of mean curvature flow. The classification problem of ancient solutions under some geometric conditions can be view as a parabolic analogue of geometric Liouville theorem. The previous results technically reply on the assumption of convexity of the curves. In the ongoing project joint with Kyeongsu Choi, Donghwi Seo, and Weibo Su, we replace it by the boundedness of entropy, which is a measure of geometric complexity defined by Colding and Minicozzi. In this talk, we will prove that an ancient smooth curve shortening flow with finite-entropy embedded in $\mathbb{R}^2$ has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with multiplity $m\geq 3$ exponentially fast in any compact region, unless the flow is a shrinking circle, a static line, a paper clip, or a translating grim reaper. In addition, we figure out the exact numbers of tips, vertices, and inflection points of the curves at negative enough time. Moreover, the exponential growth rate of graphical radius and the convergence of vertex regions to grim reaper curves will be shown.

Speaker: Tin Yau Tsang (New York University)
Title: Mass for the Large and the Small
Abstract: The positive mass theorem concerns the mass of large manifolds. In this talk, we will first review the proofs by Schoen and Yau, then the proof by Witten. Combining these with their recent generalisations turns out to help us understand the mass of small manifolds.

Speaker: Xiaolong Li (Wichita State University)
Title: Recent Developments on the Curvature Operator of the Second Kind
Abstract: In this talk, I will first introduce the curvature operator of the second kind and talk about the resolution of Nishikawa's conjecture by Cao-Gursky-Tran, myself, and Nienhaus-Petersen-Wink. Then I will talk about some ongoing research with Gursky concerning negative lower bounds of the curvature operator of the second kind. Along the way, I will mention some interesting problems.

In this talk we discuss the roles of concavity, symmetry and subsolutions in the study of fully nonlinear PDEs, especially those on real or complex manifolds with connection to geometric problems. We shall report some of our results along the line, which give the optimal conditions for the existence of classical solutions, either of the Dirichlet problem, or of equations on closed manifolds. If time permits, we shall also discuss the possibility to weaken or extend these conditions, and a class of equations involving differential forms of higher rank, more specifically real (p, p) forms for p > 1 on complex manifolds. Part of the talk is based on joint work with my student Mathew George.

Mean curvature flow is the negative gradient flow of the area
functional, and it has attracted a lot of interest in the past few years. In
this talk, we will discuss a PDE-based, gauge theoretic, construction of
codimension-two mean curvature flows based on the Yang-Mills-Higgs
functionals, a natural family of energies associated to sections and metric
connections of Hermitian line bundles. The underlying idea is to approximate
the flow by the solution of a parabolic system of equations and study the
corresponding singular limit of these solutions as the scaling parameter
goes to zero. This is based on joint work with A. Pigati and D. Stern.