The seminar usually holds on Wednesday. For more details, please visit
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Wednesday, February 22, 9:00-10:00, Zoom link
(ID: 841 8136 2356, Code: 260070)
Jingang Xiong (Beijing Normal University) - Asymptotic Analysis of Harmonic Maps With Prescribed Singularities - Abstract
Motivated by stationary vacuum solutions of the Einstein field equations, we study singular harmonic maps from domains of 3-dimensional Euclidean space to the hyperbolic plane having bounded hyperbolic distance to Kerr harmonic maps.
In the degenerate case, we prove that every such harmonic map admits a unique tangent harmonic map at the extreme black hole horizon. The possible tangent maps are classified and proved to be integrable. Expansions in the asymptotically flat end are presented.
These results, together with those of Li-Tian and Weinstein around 1990, provide a complete regularity theory for such harmonic maps prescribed singularities on $z$-axis. This is joint with Q. Han, M. Khuri and G. Weinstein.
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Wednesday, March 1, 9:00-10:00, Zoom link
(ID: 869 4880 0369, Code: 673249)
Fei He (Xiamen University) - Biharmonic heat equation on complete manifolds - Abstract
I will present some recent work on the biharmonic heat equation, which is a fourth order parabolic equation, on complete Riemannian manifolds. The results I'll talk about include estimates of the biharmonic heat kernel, a uniqueness criteria, and a uniform L-infinite estimate for solutions with bounded initial data.
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Wednesday, March 8, 9:00-10:00, Zoom link
(ID: 827 8895 6364, Code: 933760)
Sheng Rao (Wuhan University) - Deformation invariance of plurigenera of Moishezon varieties - Abstract
This talk mainly concerns deformation invariance of plurigenera, particularly on a question of Demailly whether a smooth family of nonsingular projective varieties admits the deformation invariance of plurigenera. We confirm this more generally for a smooth fiberwise Moishezon family, and also prove a flat family case.
And several more recent progresses on Kollar's and Siu's conjectures are also discussed. This talk is mainly based on several joint works with I-Hsun Tsai, Yi Li and Runze Zhang.
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Wednesday, March 15, 9:00-10:00, Zoom link
(ID: 854 4811 2768, Code: 555154)
Jingrui Cheng (Stony Brook University) - Interior W^{2,p} estimates for complex Monge-Ampere equations - Abstract
The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function.
The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.
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Wednesday, March 22, 9:00-10:00, Zoom link
(ID: 852 5271 9545, Code: 928839)
Zhihan Wang (Princeton University) - Translating mean curvature flow with prescribed end - Abstract
Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or
higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on the joint work with Ao Sun.
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Wednesday, March 29, 9:00-10:00, Zoom link
(ID: 841 6885 6498, Code: 612062)
Nick Edelen (University of Notre Dame) - A Liouville-type theorem for cylindrical cones - Abstract
It was shown by Hardt-Simon and Wang that any minimizing hypercone fits into a foliation of minimizing hypersurfaces which, apart from the cone itself, are all smooth radial graphs asymptotic to the cone.
When the cone has an isolated singularity Hardt-Simon additionally showed the foliation is unique, in the sense that any complete area-minimizing hypersurface lying to one side of the cone must be a leaf of the foliation.
In this talk we show a similar uniqueness for the foliation associated to minimizing cylindrical hypercones of the form $C = C_0 \times R^k$, when $C_0$ is smooth, strictly-minimizing, and strictly stable. This is joint work with Gábor Székelyhidi.
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Wednesday, April 12, 9:00-10:00, Zoom link
(ID: 899 0145 9675, Code: 911245)
Wenkui Du (University of Toronto) - Classification of some ancient solutions of mean curvature flow - Abstract
In this talk, we investigate the formation of singularities in mean curvature flow. Specifically, we study ancient asymptotically cylindrical flows, i.e. ancient solutions whose tangent flow at $-\infty$ is a round shrinking cylinder $\mathbb{R}^{k}\times S^{n-k}(\sqrt{2(n-k)|t|})$, where $1\leq k\leq n-1$. While in the neck case, i.e. for $k=1$, a complete classification has been obtained in several breakthroughs, a classification for the case $2\leq k\leq n-1$ until recently seemed out of reach.
To analyze ancient asymptotically cylindrical flows for $2\leq k\leq n-1$, we consider the cylindrical profile function $u$ that measures the deviation of the renormalized flow from the round cylinder. We prove that for $\tau\to -\infty$ we have the asymptotics $u(y,\omega,\tau)= (y^\top Qy -2\textrm{tr}(Q))/|\tau| + o(|\tau|^{-1})$, where $Q$ is a constant symmetric $k\times k$-matrix whose eigenvalues are quantized to be either 0 or $-\sqrt{(n-k)/8}$.
We then focus on the extremal rank cases. Under the natural noncollapsing condition, we obtain a classification of all solutions with $\textrm{rk}(Q)=0$, and establish $\textrm{SO}(n-k+1)$-symmetry and unique asymptotics in the case $\textrm{rk}(Q)=k$, also known as the $k$-oval case.
Next, we confirm a conjecture by Angenent-Daskalopoulos-Sesum about uniqueness of $\textrm{O}(k) \times \textrm{O}(n-k+1)$-symmetric ancient ovals and more generally classify all $\textrm{O}(k) \times \textrm{O}(n-k+1)$-symmetric ancient noncollapsed solutions. On the other hand, for every $2\leq k\leq n-1$ we construct a $(k-1)$-parameter family of ancient ovals that are only $\mathbb{Z}^{k}_{2}\times \mathrm{O}(n-k+1)$-symmetric, giving counterexamples to another conjecture of Daskalopoulos.
We then investigate ancient ovals without any symmetry assumption. Specifically, we prove that any $2$-oval in $\mathbb{R}^4$, up to scaling and rigid motion, either is the unique $\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by White and Haslhofer-Hershkovits, or belongs to our new one-parameter family of $\mathbb{Z}_2^2\times \textrm{O}(2)$-symmetric ancient ovals. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar.
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Wednesday, April 19, 9:00-10:00, Zoom link
(ID: 838 1058 5161, Code: 949453)
Junsheng Zhang (University of California, Berkeley) - A note on the supercritical deformed Hermitian-Yang-Mills (dHYM) equation - Abstract
We show that on a compact Kähler manifold all real (1,1)-classes admitting solutions to the supercritical dHYM equation form a both open and closed subset of those which satisfy the numerical condition proposed by Collins-Jacob-Yau.
More importantly, we show by an example that it can be a proper subset. This disproves a conjecture made by Collins-Jacob-Yau.
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Wednesday, April 26, 9:00-10:00, Zoom link
(ID: 821 5918 7625, Code: 431030)
Chao Xia (Xiamen University) - Heintze-Karcher inequality and Alexandrov's theorem for capillary hypersurfaces - Abstract
Heintze-Karcher's inequality is an optimal geometric inequality for embedded closed hypersurfaces, which can be used to prove Alexandrov's soap bubble theorem on embedded closed CMC hypersurfaces in the Euclidean space.
In this talk, we introduce a Heintze-Karcher-type inequality for hypersurfaces with boundary in the half-space. As application, we give a new proof of Wente's Alexandrov-type theorem for embedded CMC capillary hypersurfaces.
Moreover, the proof can be adapted to the anisotropic case, which enable us to prove an Alexandrov-type theorem for embedded anisotropic capillary hypersurfaces. This is based on joint works with Xiaohan Jia, Guofang Wang and Xuwen Zhang.
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Wednesday, May 10, 9:00-10:00, Zoom link
(ID: 839 1359 1420, Code: 051649)
Shuli Chen (Stanford University) - A Generalization of the Geroch Conjecture with Arbitrary Ends - Abstract
The Geroch conjecture (proven by Schoen-Yau and Gromov-Lawson) says that the torus T^n does not admit a metric of positive scalar curvature. In this talk, I will explain how to generalize it to some non-compact settings using μ-bubbles.
In particular, I will talk about why the connected sum of a Schoen-Yau-Schick n-manifold with an arbitrary n-manifold does not admit a complete metric of positive scalar curvature for n <=7; this generalizes work of Chodosh and Li.
I will also discuss about how to generalize Brendle-Hirsch-Johne's non-existence result for metrics of positive m-intermediate curvature on N^n = M^{n-m} x T^m to to manifolds with arbitrary ends for n <= 7 and certain m.
Here, m-intermediate curvature is a new notion of curvature interpolating between Ricci and scalar curvature.
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Wednesday, May 17, 9:00-10:00, Zoom link
(ID: 834 7812 4316, Code: 498338)
Huaiyu Zhang (Nanjing University of Science & Technology) - Positive mass theorem for metrics with singular sets - Abstract
The classical positive mass theorem compares the ADM mass of an asymptotically flat manifold with non-negative scalar curvature to it of the Euclidean space. Inspired by Gromov's series papers and Schoen's conjecture,
people have great intentions to study the positive mass theorem for metrics with singular sets. In this talk, I will talk about our recent work on this topic.
This is a joint work with Wenshuai Jiang and Weimin Sheng.
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Wednesday, May 24, 9:00-10:00, Zoom link
(ID: 892 6580 0195, Code: 607196)
Xieping Wang (University of Science and Technology of China) - Complete pluripolar sets and removable singularities of plurisubharmonic functions - Abstract
We will talk about a Hartogs type extension theorem for psh functions across a compact complete pluripolar set. To put this result in a historical context, we will also review several classical results on psh functions and closed positive currents.
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Wednesday, May 31, 9:00-10:00, Zoom link
(ID: 811 0028 7715, Code: 165951)
Yevgeny Liokumovich (University of Toronto) - Parametric geometric inequalities and their applications - Abstract
I will talk about parametric versions of two classical inequalities in geometry: the isoperimetric inequality and coarea inequality. These two inequalities are basic tools in analyzing properties of submanifolds in Riemannian geometry. The parametric versions are statements about a family of submanifolds as opposed to one single submanifold.
I will discuss proofs of some parametric inequalities in low dimensions, the higher dimensional versions that are still open and their applications to minimal surfaces and stationary geodesic nets. The talk will be based on joint works with Larry Guth and Bruno Staffa.
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Wednesday, June 14, 9:00-10:00, Zoom link
(ID: 878 8399 5957, Code: 972661)
Yuanyuan Lian (Shanghai Jiao Tong University) - Pointwise Regularity for Fully Nonlinear Elliptic Equations in General Forms - Abstract
In this talk, we develop systematically the pointwise regularity for viscosity solutions of fully nonlinear elliptic equations in general forms. In particular, the equations with quadratic growth (called natural growth) in the gradient are covered. We obtain a series of interior pointwise $C^{k,\alpha}$ regularity ($k\geq 1$ and $0<\alpha<1$).
Some regularity results are new even for the linear equations. Moreover, the minimum requirements are imposed to obtain above regularity and our proofs are simple.