Geometric Analysis Seminar
Organizers
Pak Tung Ho (Tamkang University)
Jinwoo Shin (Sookmyung Women's University)
Please contact [shinjin "at" sookmyung "dot" ac "dot" kr] to subscribe to the mailing list for the seminar.
Upcoming Seminars
Past Seminars
2023. 11. 08 (Wed) 14:00 - 15:00 (GMT +9), online (Zoom ID : 858 4587 9055, passcode : 775372)
Manh Tien NGUYEN (Université Libre de Bruxelles)
Title : Gradient Estimates for a Weighted Nonlinear Parabolic Equation under Weighted Yamabe Flow and Some Applications
Abstract : Gradient estimates for positive solutions of nonlinear parabolic equations have a long history, which dates back to the works of Peter Li, Shin Tung Yau, and Hamilton in the 1980s. They have numerous applications in the field of geometric analysis, in particular, the convergence of geometric flows. Many results on different types of gradient estimates have been studied, which generalized the original works and concerned different geometric settings. In this talk, we will see the gradient estimates of a weighted parabolic equation under the weighted Yamabe flow. We also mention Harnack-type inequalities, a second-order gradient estimate, and the monotonicity of parabolic frequency, which is of great interest at present, as a consequence of the gradient estimates.
2023. 09.26 (Tue) 15:00 - 16:00 (GMT +9), online (Zoom ID : 815 4162 8066, passcode : 388175)
Jialong Deng (Tsinghua University )
Title : The weighted scalar curvature
Abstract : Inspired by the importance of the Bakry-Emery curvature on a weighted Riemannian manifold $(M^n, g, e^{f}Vol_g)$, we will introduce the weighted scalar curvature on it and then extend some classic results of the scalar curvature to the weighted version. For example, we will generalize Schoen-Yau's minimal hypersurface method, Gromov-Lawson's index theory approach and Seiberg-Witten invariants (in four dimensions) to a weighted Riemannian manifold. General relativity with the weighted scalar curvature and the prescribing weighted scalar curvature problem will mention, if time permits.
2023. 01. 12 (Thu) 11:00 - 12:00 (GMT +9), online (Zoom ID: 886 7903 4976
passcode: 269913)
Tang-Kai Lee (MIT)
Title : Parabolic frequency on geometric flows
Abstract : Frequency functions were invented by Almgren to study the regularities of mass-minimizing currents. Their monotonicity properties have several applications in geometric analysis. A parabolic analog of Almgren's frequency was defined and investigated by Poon, and extended to the manifold setting later. In this talk, we will see how parabolic frequency functions are used to study the solutions to the heat equation along geometric flows. This is joint work with Julius Baldauf and Pak Tung Ho.
2022. 12. 08 (Thu) 17:00 - 18:00 (GMT +9), KIAS
Paul-Andi Nagy (IBS-CCG)
Title : Eigenvalue estimates for 3-Sasaki structures
Abstract : We obtain new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian for 3-Sasaki metrics, improving Lichnerowicz-Obata type estimates by Ivanov et al. The limiting eigenspace is fully described in terms of the automorphism algebra. Our results can be thought of as an analogue of the Lichnerowicz-Matsushima estimate for Kähler-Einstein metrics. In dimension 7, if the automorphism algebra is non-vanishing, we also compute the second eigenvalue for the sub-Laplacian and construct explicit eigenfunctions. In addition, for all metrics in the canonical variation of the 3-Sasaki metric we give a lower bound for the spectrum of the Riemannian Laplace operator, depending only on scalar curvature and dimension. We also strengthen a result pertaining to the growth rate of harmonic functions, due to Conlon, Hein and Sun, in the case of hyperkähler cones. In this setup we also describe the space of holomorphic functions.
2022. 12. 29 (Thu) 11:00 - 12:00 (GMT +9), online (Zoom ID : 847 9880 5715, passcode : 456218)
Zhehui Wang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Title : Global behaviors of minimal graphs over unbounded convex domains
Abstract: In this talk, we will study global behaviors of minimal graphs over unbounded convex domains. We will first recall a Liouville type theorem for the minimal surface equation in the half space with constant Neumann boundary value. Then, we will recall a Bernstein type theorem in unbounded convex domains in any dimension. Finally, we will prove a stability type theorem which is related to it. This talk is based on joint works with Nick Edelen, and with Guosheng Jiang and Jintian Zhu.
2022. 11. 17 (Thu) 17:00 - 18:00 (GMT +9), online (Zoom ID : 882 3188 3710, passecode : 689771)
Mingxiaong Li (Nanjing University)
Title : Some existence results of semilinear equations with sign-changing potentials
Abstract : We study the existence of the solutions to mean field equations with critical values. A sufficient condition is given with sign-changing potentials. This is based on the joint work with Xingwang Xu. Moreover, we generalize the result to higher order cases with help of similar ideas.
2022. 11. 10 (Thu), 11:00 - 12:00 (GMT +9), online(Zoom ID : 839 4240 2150, passcode : 724805)
Kuang-Ru Wu (Institute of Mathematics, Academia Sinica)
Title : Positively curved Finsler metrics on vector bundles
Abstract : While the equivalence between ampleness and positivity holds for vector bundles of rank one, its higher rank counterpart known as Griffiths' conjecture is still open. There is also a similar but weaker conjecture by Kobayashi who proposed to use Finsler rather than Hermitian metrics to study the equivalence. We will review these two conjectures and state our progress. One of our results is that Kobayashi positivity implies ampleness and convex Kobayashi positivity. We will also discuss how to prove Kobayashi positivity for ample vector bundles with additional curvature assumptions.
2022. 11. 03 (Thu), 11:00 - 12:00 (GMT +9), online(Zoom ID : 865 8619 3257, passcode : 384932)
Ting-Jung Kuo (National Taiwan Normal University)
Title : Curvature equation with conic singularities, integrable system and complex ODEs
Abstract : pdf
2022. 10. 27 (Thu), 11:00 - 12:00 (GMT +9), online(Zoom ID : 810 8548 6340, passcode : 270280)
Chanyoung Sung (Korea National University of Education)
Title : Some lower bounds of functionals in geometric variational problems
Abstract : We first establish a framework for fiberwise spherical or planar symmetrization to find a lower bound of a Dirichlet-type energy functional in a variational problem on a fibred Riemannian manifold, and use it to obtain lower bounds of the first eigenvalue of Laplacian and the Yamabe constant on a fibered Riemannian manifold. Secondly we give a lower bound of the Weyl functional on a 4-manifold with a metric of positive Yamabe constant, which generalizes Gursky's inequality.
2022. 10. 06 (Thu) 11:00 - 12:00 (GMT +9), online (Zoom ID : 870 8531 9915, passcode : 063335)
Eric Chen (University of California, Berkeley)
Title : The Yamabe flow on asymptotically Euclidean manifolds
Abstract : In most cases on compact manifolds, the Yamabe flow converges to a metric of constant scalar curvature. But on noncompact manifolds, long-time existence or convergence may fail. I will discuss the behavior of the flow on asymptotically Euclidean manifolds. In this case, long-time existence of the Yamabe flow always holds, and the flow converges if and only if the Yamabe constant of the initial metric's conformal class is positive. When convergence fails in the case of a nonpositive Yamabe constant, the blowup profile at time infinity can be described using the solution of the Yamabe problem on a singular compactification of the original manifold. This is joint work with Gilles Carron and Yi Wang.
2022. 09. 29 (Thu) 11:00 - 12:00 (GMT +9), online (Zoom ID : 857 8423 6342, passcode : 295431)
Chengjie Yu (Shantou University)
Title : Hadamard's three circle theorem revisited
Abstract : In this talk, we will present our recent work extending Gang Liu's three circle theorem which is an natural extension of Hadamard's classical theorem, and give some applications. This is a joint work with Chuangyuan Zhang.
2022. 07. 28 (Thu) 17:00 - 18:00 (GMT +9), KIAS (in person)
Gabjin Yun (Myongji University)
Title : Vacuum static spaces with positive isotropic curvature
Abstract : pdf
2022. 07. 21 (Thu) 17:00 - 18:00 (GMT +9), KIAS (in person)
Juncheol Pyo (Pusan National University)
Title : Sobolev inequality and isoperimetric inequality for submanifolds in a smooth metric measure space
Abstract : Brendle recently proved a sharp Sobolev inequality and logarithmic Sobolev inequality for submanifolds in Euclidean space.
From the sharp Sobolev inequality, he achieved a breakthrough on the conjecture of isoperimetric inequality for minimal submanifolds.
In this talk, we extend the Brendle's results to submanifolds in a smooth metric measure space. As an application, we prove some new isoperimetric-type inequalities in some smooth metric measure spaces. For example, we obtain a new isoperimetric-type inequality for the self-expander. This is joint work with Pak-Tung Ho.
2022. 07. 14 (Thu) 17:00 - 18:00 (GMT +9), Online (Zoom ID : 870 4159 3788, passcode : 200487)
Luke Peachey (Warwick University)
Title : Non-uniqueness of curve shortening flow
Abstract : One can view curve shortening flow, or more generally mean curvature flow, as a geometric or non-linear version of the heat equation. In 1935, Tychonoff showed a non-uniqueness result for the heat equation on Euclidean space when you allow solutions to have sufficiently large growth at infinity. It remains an interesting open problem if non-closed curves admit a unique evolution under curve shortening flow within a suitable class of solutions. I will first formulate such a uniqueness conjecture in Euclidean space, before presenting a result analogous to Tychonoff’s for curve shortening flow in more general ambient surfaces, where we instead allow the curvature of our ambient surface to have sufficiently large growth at infinity.
2022. 07. 07 (Thu) 15:00 - 15:40 (GMT +9), KIAS 1423(in person)
Hanna Kim (University of Illinois at Urbana-Champaign )
Title : Maximization of the second Laplacian eigenvalue on the sphere
Abstract : Isoperimetric problem for eigenvalues of the Laplacian on a given manifold is concerned with finding an upper bound and understands for which metrics the upper bound is attained. On the sphere, the first nonzero eigenvalue can be maximized by the standard “round” metric. For 2-dimensional sphere, the second eigenvalue becomes maximal as the surface degenerates to two disjoint spheres by a result of Nadirashvili. It was conjectured by Girouard, Nadirashvili and Polterovich that for higher dimensional spheres, the second eigenvalue has an analogous upper bound. In this presentation, I will establish a proof by construction of trial functions confirming the conjecture.
2022. 07. 07 (Thu) 17:00 - 18:00 (GMT +9), Online (Zoom ID : 868 8519 1857, passcode : 948544)
Tiarlos Cruz (Universidade Federal de Alagoas )
Title : Existence and obstructions for the curvature on manifolds with boundary
Abstract : A theme of long standing interest concerns the study of the set of curvature functions which a given compact and non-compact manifold with nonempty boundary can possess. We show that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the geodesic curvature or the Gaussian curvature of some conformally equivalent metric. We also discuss new existence and nonexistence of metrics with prescribed curvature in the conformal setting which depends on the Euler characteristic. After this, we present a higher order analogue concerning scalar and mean curvatures on compact manifolds with boundary. We also give conditions for Riemannian manifolds not necessarily complete or compact to admit positive scalar curvature and minimal boundary. This talk is based on joint works with F. Vitório and A. Santos.
2022. 06. 07. (Tue) 17:00 - 18:00 (GMT +9), Online (Zoom ID : 897 2526 2792, passcode : 071016)
Romain Petrides (Université de Paris)
Title : Minimizing combinations of Laplace eigenvalues and applications
Abstract : We give a variational method for existence and regularity of metrics which minimize combinations of eigenvalues of the Laplacian among metrics of unit area on a surface. We show that there are minimal immersions into ellipsoids parametrized by eigenvalues, such that the coordinate functions are eigenfunctions with respect to the minimal metrics.
As one of the applications, we explain a new method to construct non- planar minimal spheres into 3d-ellipsoids after Haslhofer-Ketover and Bettiol-Piccione.
2022. 06. 09 (Thu) 17:00 - 18:00 (GMT +9), KIAS (in person)
Keomkyo Seo (Sookmyung Women's University)
Title : Free boundary constant mean curvature surfaces in a convex domain
Abstract : We discuss a rigidity theorem for free boundary constant mean curvature surfaces in a convex domain in a 3-dimensional Riemannian manifold with sectional curvature bounded above by a constant, which states that such surface is homeomorphic to either a disk or an annulus under an appropriate pinching condition on the length of the traceless second fundamental form on the surface. This is joint work with Sung-Hong Min.
2022. 06. 02 (Thu) 11:00 - 12:00 (GMT +9), Online (Zoom ID : 863 0561 8921, passcode : 595855)
Li Ma (University of Science and Technology Beijing)
Title : Porous-media flow and Yamabe flow on bounded domains
Abstract : In this talk, we discuss Yamabe (porous media) flow on a bounded regular domain $\Om$ with smooth boundary in the Euclidean space $R^n$ of dimension $n\geq 1$. After reviewing the background material, we consider the local and global flow to the flow. The gradient estimate of positive smooth solutions is presented and the normalization process of the flow is also concerned.
2022. 05. 26. (Thu) 17:15 - 18:15 (GMT +9), Online (Zoom ID : 897 7228 8315, passcode : 846108)
Wei Yuan (Sun Yat-Sen University)
Title : Area comparison of hypersurfaces in space forms
Abstract : Mean curvature is one of the most fundamental extrinsic curvature in the theory of submanifold. A natural question is that wether mean curvature can control the area of hypersurfaces. In this talk, we discuss the area comparison with respect to mean curvature for hypersurfaces in space forms. This is a joint work with Professor Sun Jun in Wuhan University.
2022. 05. 12. (Thu) 10:00 - 11:00 (GMT +9), Online (Zoom ID : 852 6835 6621, passcode : 452338)
Guillermo Henry (Universidad de Buenos Aires)
Title : Solutions of the Yamabe equation induced by subgroups of isometries and isoparametric functions
Abstract : Given a closed $n-$dimensional Riemannian manifold $(M,g)$ with scalar curvature $s_g$. We say that $u$ is a solution of the Yamabe equation if for some constant $c$ it holds $$\frac{4(n-1)}{(n-2)}\Delta_g u+s_gu=c|u|^{\frac{4}{n+2}}u.$$
The solutions of this equation are interesting because they are related with metrics of constant scalar curvature in the conformal class of $g$.
In this talk we are going to discuss some results on the existence of positive and sign-changing solutions of the Yamabe equation obtained by considering either the action of a subgroup of isometries or an isoparametric foliation.
2022. 04. 28. (Thu) 10:00 - 11:00 (GMT +9), Online (Zoom ID : 828 9090 9400, passcode : 326004)
Zetian Yan (Penn State University)
Title : Improved higher-order Sobolev inequalities on CR sphere.
Abstract : We improve higher order CR Sobolev inequalities on S^{2n+1} under the
vanishing of higher order moments of the volume element. As an application, we give a new and direct proof of the classification of minimizers of the CR invariant higher-order Sobolev inequalities. In the same spirit, we prove almost sharp Sobolev inequalities for GJMS operators to general CR manifolds, and obtain existence of minimizers in C^{2k}(N) of high-order CR Yamabe-type problems when Y_k(N)<Y_k(H^n).
2022. 04. 21. (Thu) 11:00 - 12:00 (GMT +9), Online (Zoom ID : 841 6087 8263, passcode : 134927)
Liming Sun (Academy of Mathematics and Systems Science)
Title : Quantitative stability of harmonic maps from R^2 to S^2 with higher degree.
Abstract : For degree 1 harmonic maps from R^2 (or S^2) to S^2, Bernand-Mantel, Muratov and Simon recently establish a uniformly quantitative stability estimate. Namely, for any map u from R^2 to S^2, the discrepancy of its Dirichlet energy and 4pi can linearly control the H^1-difference of u from the set of degree 1 harmonic maps. Whether a similar estimate holds for harmonic maps with higher degree is unknown. In this paper, we prove that a similar quantitative stability result for higher degree is true only in local sense. Namely, given a harmonic map, a similar estimate holds if it is already sufficiently near to it (modulo Mobius transforms) and the bound in general depends on the given harmonic map. More importantly, we investigate an example of degree 2 case thoroughly, which shows that it fails to have a uniformly quantitative estimate like the degree 1 case. This phenomenon show the striking difference of degree 1 ones and higher degree ones. Finally, we also conjecture a new uniformly quantitative stability estimate based on our computation. This work is joint with Bin Deng and Jun-cheng Wei.
2022. 04. 07. (Thu) 17:00 - 18:00, Online (Zoom ID : 829 3960 9101, passcode : 455740)
Quoc Anh Ngo (Vietnam National University)
Title : On the Hang-Yang conjecture for GJMS equations on S^n
Abstract : In this talk, I will describe a new approach to obtain Liouville-type results for positive smooth solutions to a linearly perturbed critical/subcritical GJMS equation on S^n. Here the underlying GJMS operator is of order 2m, which is greater than n. In the special when m=2 and n=3, such a Liouville-type result was conjectured by F. Hang and P. Yang in a recent work published in Math. Res. Not. IMRN. As a by-product, I will explain how to obtain sharp critical/subcritical Sobolev inequalities for GJMS operator on S^n. This is joint work with Ali Hyder.
2022. 3. 31. (Thu) 17:00 - 18:00, Online (Zoom ID : 816 7711 8169, passcode : 975921)
Daniele Angella (Università di Firenze)
Title : The Hermitian geometry of the Chern connection
Abstract : We consider some problems concerning the geometry of the Chern connection of Hermitian manifolds, e.g.: the existence of metrics with constant Chern-scalar curvature; the generalizations of the Kahler-Einstein condition to the non-Kahler setting; the convergence of the normalized Chern-Ricci flow on compact complex surfaces.
The talk is based on collaborations and discussions with Simone Calamai, Francesco Pediconi, Cristiano Spotti, Valentino Tosatti.
2022. 3. 24. (Thu) 10:00 - 11:00, Online (Zoom ID : 833 1527 0187, passcode : 245129)
Cheikh Birahim Ndiaye (Howard University)
Title : A Complete Variational theory for Yamabe type problems.
Abstract : In this talk, we will discuss a Complete Variational theory for Yamabe type problems. We will show how the combination of the Aubin-Schoen’s Minimizing argument and the Bahri-Coron’s Barycenter technique leads to a natural solution for such problems without any Positive Mass type assumption. To emphasize the main ideas of the theory, we will focus on the High-order Fractional Yamabe problem on Poincare-Einstein manifolds discussing a joint work with Y. Sire and L. Sun.
2022. 3. 17 (Thu) 10:00 - 11:00, Online (Zoom ID : 842 8355 9637, passcode : 824226)
Sergio Almaraz (Universidade Federal Fluminense)
Title : A singular Yamabe problem on manifolds with solid cones.
Abstract : We study the existence of conformal metrics on non-compact Riemannian manifolds with non-compact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d>(n-2)/2. Our main theorem is inspired by the classical results by Aviles-McOwen and Loewner-Nirenberg known in the literature as the “singular Yamabe problem”. This is a joint work with Juan Pablo Alcon (UFF).
2022. 3. 10. (Thu) 11:00 - 12:00, Online (Zoom Id : 851 4142 8837, passcode : 262207)
Fang Wang (Shanghai Jiao Tong University)
Title : The relative volume of Poincare-Einstein manifolds
Abstract: For a Poincare-Einstein manifold, the Bishop-Gromov comparison theorem tells us that the relative volume is a non-increasing function of the geodesic radius. In this talk, I will show that the fractional Yamabe constant at the conformal infinity provides a lower bound for this function. As an application, this implies a gap phenomena and the rigidity theorem.
2022. 3. 3. (Thu) 11:00 - 12:00, Online (Zoom ID : 827 3426 4600, passcode : 858876)
Hong Zhang (South China Normal University)
Title : Evolution of scalar curvature flow on S^n to a prescribed sign-changing function
Abstract : The talk focuses on the problem of prescribing scalar curvature on n-sphere, the so-called Nirenberg problem. A widely used assumption in the existed works is that the prescribed function f is of positive sign. In this talk, we relax this restriction to allow f to change sign. We use the well-known scalar curvature flow to study the problem. We prove that the scalar curvature flow converges, as time goes to infinity, to a conformal metric having f as its scalar curvature provided that f satisfies some Morse index counting condition or symmetry condition. As direct consequences, various existence theorems can be derived for the Nirenberg problem.
2022. 2. 17. (Thu) 17:00 - 18:00, Online(Zoom ID : 852 3587 7670, passcode : 747545)
Hardy Chan (Instituto de Ciencias Matemáticas)
Title : Local and nonlocal ODEs in the singular fractional Yamabe problem
Abstract : In conformal geometry, the Yamabe problem asks for Yamabe metrics, or conformal metrics of constant scalar curvature. In search of singular Yamabe metrics, one is led to the study of the Lane-Emden equation with a Sobolev-subcritical (or -critical) exponent that depends on the dimension of the singularity. The radial profile, which solves a classical ODE, is well-understood.
One could pose the same problem concerning the fractional curvature, a general notion that includes the scalar curvature, the curvatures associated with Paneitz and GJMS operators, as well as those with non-integer order. For the investigation of the corresponding radial profile, we discuss the development of the nonlocal ODE theory. Apart from the localizing Caffarelli-Silvestre extension, we show that nonlocal ODE can also be understood as a coupled infinite system of second order ODEs. Finally, we also mention a simple while surprising transformation that reduces the nonlocal ODE into almost a scalar first order ODE.