Department of Mathematics - Geometric Analysis and Application
https://math.yale.edu/seminars/geometric-analysis-and-application
enMin-max free boundary minimal surface with genus at least one
https://math.yale.edu/event/min-max-free-boundary-minimal-surface-genus-least-one
<div class="label-inline">Time: </div><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2022-04-29T14:00:00-04:00">2:00pm</span><div class="label-above">Speaker: </div>Yuchin Sun <div class="label-above">Abstract: </div><div class="tex2jax"><p>Abstract:</p>
<p>We build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus at least one and with boundary. We show that the width for the area functional can be achieved by a bubble tree limit consisting of branched genus g free boundary minimal surfaces with nodes, possibly finitely many branched minimal spheres, and free boundary minimal disks. This provides an existence result for free boundary minimal surfaces with genus of arbitrary codimension.</p>
</div><div class="label-inline">Seminar: </div><a href="/seminars/geometric-analysis-and-application">Geometric Analysis and Application</a><div class="label-above">Undergraduate Event?: </div>FalseFri, 22 Apr 2022 14:03:15 +0000Amy Davis24508 at https://math.yale.eduhttps://math.yale.edu/event/min-max-free-boundary-minimal-surface-genus-least-one#commentsConvergence of the self-dual U(1)-Yang-Mills-Higgs energies to the (n - 2)-area functional
https://math.yale.edu/event/convergence-self-dual-u1-yang-mills-higgs-energies-n-2-area-functional
<div class="label-inline">Time: </div><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2022-04-22T14:00:00-04:00">2:00pm</span><div class="label-above">Speaker: </div>Davide Parise <div class="label-above">Abstract: </div><div class="tex2jax"><p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>Abstract: </span></p>
<p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>We overview the recently developed level set approach to the existence theory of minimal submanifolds and present some joint work with A. Pigati and D. Stern. The underlying idea is to construct minimal hypersurfaces as limits of nodal sets of critical points of functionals. After starting with a general overview of the codimension one theory, we will move to the higher codimension setting, and introduce the self-dual Yang-Mills-Higgs functionals. These are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points have long been studied in gauge theory. We will explain to what extent the variational theory of these energies is related to the one of the (n - 2)-area functional and how one can interpret the former as a relaxation/regularization of the latter. We will mention some elements of the proof, with special emphasis on the role played by the gradient flow.</span></p>
</div><div class="label-inline">Seminar: </div><a href="/seminars/geometric-analysis-and-application">Geometric Analysis and Application</a><div class="label-above">Undergraduate Event?: </div>FalseMon, 18 Apr 2022 11:42:28 +0000Amy Davis24506 at https://math.yale.eduhttps://math.yale.edu/event/convergence-self-dual-u1-yang-mills-higgs-energies-n-2-area-functional#commentsRotational Symmetry of Mean Curvature Flows coming out of a double cone
https://math.yale.edu/event/rotational-symmetry-mean-curvature-flows-coming-out-double-cone
<div class="label-inline">Time: </div><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2022-03-11T14:00:00-05:00">2:00pm</span><div class="label-above">Speaker: </div>Letian Chen<div class="label-above">Abstract: </div><div class="tex2jax"><p>Abstract:</p>
<p>We show that any initially smooth mean curvature flow (MCF) coming out of a rotationally symmetric double cone, which is singular at the origin, must stay rotationally symmetric for all time. Examples of such flows include (smooth) self-expanders, which are potential models of continuations of MCFs through conical singularities. I will also discuss non-self-similar, possibly singular, MCFs coming out of a cone that fit into our theorem</p>
</div><div class="label-inline">Seminar: </div><a href="/seminars/geometric-analysis-and-application">Geometric Analysis and Application</a><div class="label-above">Undergraduate Event?: </div>FalseFri, 04 Mar 2022 21:31:46 +0000Amy Davis24478 at https://math.yale.eduhttps://math.yale.edu/event/rotational-symmetry-mean-curvature-flows-coming-out-double-cone#commentsKähler-Einstein metrics on complex hyperbolic cusps
https://math.yale.edu/event/kahler-einstein-metrics-complex-hyperbolic-cusps
<div class="label-inline">Time: </div><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2022-05-06T14:00:00-04:00">2:00pm</span><div class="label-above">Speaker: </div>Hans-Joachim Hein <div class="label-above">Abstract: </div><div class="tex2jax"><p><span>Abstract: </span></p>
<p><span>A complex hyperbolic cusp is an end of a finite-volume quotient of complex hyperbolic space. Up to a finite cover, any such cusp can be realized as the punctured unit disk bundle of a negative line bundle over an abelian variety. The Dirichlet problem for complete Kähler-Einstein metrics on this space with boundary data prescribed on the unit circle bundle is well-posed. We determine the precise asymptotics of its solutions towards the zero section. Time permitting I will also mention an application to gluing constructions for Kähler-Einstein metrics on surfaces of general type. This is joint work with Xin Fu and Xumin Jiang.</span></p>
</div><div class="label-inline">Seminar: </div><a href="/seminars/geometric-analysis-and-application">Geometric Analysis and Application</a><div class="label-above">Undergraduate Event?: </div>FalseWed, 12 Jan 2022 19:40:16 +0000Amy Davis24387 at https://math.yale.eduhttps://math.yale.edu/event/kahler-einstein-metrics-complex-hyperbolic-cusps#commentsGeneralizations of Tachibana's theorem
https://math.yale.edu/event/generalizations-tachibanas-theorem
<div class="label-inline">Time: </div><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2022-04-15T14:00:00-04:00">2:00pm</span><div class="label-above">Speaker: </div>Matthias Wink <div class="label-above">Abstract: </div><div class="tex2jax"><p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>Abstract: </span></p>
<p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>A famous theorem of Tachibana says that compact Einstein manifolds with positive curvature operators have constant curvature. In this talk we will discuss several generalizations of this theorem. For example, we show that it suffices to assume that the curvature operator is $\lfloor \frac{n-1}{2} \rfloor$-positive, where $n$ is the dimension of the manifold. Time permitting, we discuss analogues of Tachibana’s theorem for K"ahler manifolds and quaternion K"ahler manifolds. This talk is based on joint work with Peter Petersen.</span></p>
</div><div class="label-inline">Seminar: </div><a href="/seminars/geometric-analysis-and-application">Geometric Analysis and Application</a><div class="label-above">Undergraduate Event?: </div>FalseWed, 12 Jan 2022 19:38:13 +0000Amy Davis24386 at https://math.yale.eduhttps://math.yale.edu/event/generalizations-tachibanas-theorem#commentsHessian Estimates for the Lagrangian mean curvature equation
https://math.yale.edu/event/hessian-estimates-lagrangian-mean-curvature-equation
<div class="label-inline">Time: </div><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2022-04-08T14:00:00-04:00">2:00pm</span><div class="label-above">Speaker: </div>Arunima Bhattacharya <div class="label-above">Abstract: </div><div class="tex2jax"><p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>Abstract: </span></p>
<p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>In this talk, we will derive a priori interior Hessian estimates for the Lagrangian mean curvature equation under certain natural restrictions on the Lagrangian phase. As an application, we will use these estimates to solve the Dirichlet problem for the Lagrangian mean curvature equation with continuous boundary data on a uniformly convex, bounded domain.</span></p>
</div><div class="label-inline">Seminar: </div><a href="/seminars/geometric-analysis-and-application">Geometric Analysis and Application</a><div class="label-above">Undergraduate Event?: </div>FalseWed, 12 Jan 2022 19:35:55 +0000Amy Davis24385 at https://math.yale.eduhttps://math.yale.edu/event/hessian-estimates-lagrangian-mean-curvature-equation#commentsTangent Flows of Kähler Metric Flows
https://math.yale.edu/event/tangent-flows-kahler-metric-flows
<div class="label-inline">Time: </div><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2022-04-01T14:00:00-04:00">2:00pm</span><div class="label-above">Speaker: </div>Max Hallgren <div class="label-above">Abstract: </div><div class="tex2jax"><p>Abstract:</p>
<p>In this talk, we will discuss some additional structure in the Kähler setting for Bamler’s limit spaces of noncollapsed Ricci flows. We will review various notions of singular set stratification, and then state an improved dimension estimate for odd-dimensional strata of limits of Kähler-Ricci flows. We also show that tangent flows of Kähler metric flows admit natural isometric actions, which are locally free away from the vertex in the case that the tangent flows are static. </p>
</div><div class="label-inline">Seminar: </div><a href="/seminars/geometric-analysis-and-application">Geometric Analysis and Application</a><div class="label-above">Undergraduate Event?: </div>FalseWed, 12 Jan 2022 19:33:09 +0000Amy Davis24384 at https://math.yale.eduhttps://math.yale.edu/event/tangent-flows-kahler-metric-flows#commentsGravitational instantons, rational surfaces, and K3 surfaces
https://math.yale.edu/event/gravitational-instantons-rational-surfaces-and-k3-surfaces
<div class="label-inline">Time: </div><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2022-03-04T14:00:00-05:00">2:00pm</span><div class="label-above">Speaker: </div>Jeff Viaclovsky <div class="label-above">Abstract: </div><div class="tex2jax"><p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>Abstract:</span></p>
<p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>I will describe some examples of complete non-compact Ricci-flat metrics in dimension 4, which are called “gravitational instantons.” In many cases, these can be compactified complex analytically to rational surfaces. I will then discuss how these gravitational instantons can arise from sequences of degenerating Ricci-flat metrics on the compact K3 surface, through a process called “bubbling”.</span></p>
</div><div class="label-inline">Seminar: </div><a href="/seminars/geometric-analysis-and-application">Geometric Analysis and Application</a><div class="label-above">Undergraduate Event?: </div>FalseWed, 12 Jan 2022 19:29:31 +0000Amy Davis24383 at https://math.yale.eduhttps://math.yale.edu/event/gravitational-instantons-rational-surfaces-and-k3-surfaces#commentsMetric SYZ conjecture
https://math.yale.edu/event/metric-syz-conjecture
<div class="label-inline">Time: </div><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2022-02-25T14:00:00-05:00">2:00pm</span><div class="label-above">Speaker: </div>Yang Li <div class="label-above">Abstract: </div><div class="tex2jax"><p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>Abstract: </span></p>
<p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>One possible interpretation of the SYZ conjecture is that for a polarized family of CY manifolds near the large complex structure limit, there is a special Lagrangian fibration on the generic region of the CY manifold. Generic here means a set with a large percentage of the CY measure, and the percentage tends to 100% in the limit. I will discuss some recent progress on this version of the SYZ conjecture, with some emphasis on the special case of the Fermat family.</span><span></span></p>
</div><div class="label-inline">Seminar: </div><a href="/seminars/geometric-analysis-and-application">Geometric Analysis and Application</a><div class="label-above">Undergraduate Event?: </div>FalseWed, 12 Jan 2022 19:27:59 +0000Amy Davis24382 at https://math.yale.eduhttps://math.yale.edu/event/metric-syz-conjecture#commentsMinimal hypersurfaces in a generic 8-dimensional closed manifold
https://math.yale.edu/event/minimal-hypersurfaces-generic-8-dimensional-closed-manifold
<div class="label-inline">Time: </div><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2022-02-18T14:00:00-05:00">2:00pm</span><div class="label-above">Speaker: </div>Yangyang Li <div class="label-above">Abstract: </div><div class="tex2jax"><p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>Abstract: </span></p>
<p style="margin-top:9.0pt; margin-right:0in; margin-bottom:0in; margin-left:0in"><span>In the recent decade, the Almgren-Pitts min-max theory has advanced the existence theory of minimal surfaces in a closed Riemannian manifold $(M^{n+1}, g)$. When $2 \leq n+1 \leq 7$, many properties of these minimal hypersurfaces (geodesics), such as areas, Morse indices, multiplicities, and spatial distributions, have been well studied. However, in higher dimensions, singularities may occur in the constructed minimal hypersurfaces. This phenomenon invalidates many techniques helpful in the low dimensions to investigate these geometric objects. In this talk, I will discuss how to overcome the difficulty in a generic 8-dimensional closed manifold, utilizing various deformation arguments. En route to obtaining generic results, we prove the generic regularity of minimal hypersurfaces in dimension 8. This talk is partially based on joint works with Zhihan Wang.</span><span></span></p>
</div><div class="label-inline">Seminar: </div><a href="/seminars/geometric-analysis-and-application">Geometric Analysis and Application</a><div class="label-above">Undergraduate Event?: </div>FalseWed, 12 Jan 2022 19:09:22 +0000Amy Davis24381 at https://math.yale.eduhttps://math.yale.edu/event/minimal-hypersurfaces-generic-8-dimensional-closed-manifold#comments