S. K. Donaldson
This paper is a discussion of relations between some free-boundary problems and infinite dimensional Lie groups; particularly a version of Nahm's equations for the group of Hamiltonian diffeomorphisms in two dimensions.
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The Nahm pole boundary condition for certain gauge theory equations in four and five dimensions is defined by requiring that a solution should have a specified singularity along the boundary. In the present paper, we show that this boundary condition is elliptic and has regularity properties analogous to more standard elliptic boundary conditions. We also establish a uniqueness theorem for the solution of the relevant equations on a half-space with Nahm pole boundary conditions. These results are expected to have a generalization involving knots, with applications to the Jones polynomial and K…
It is conjectured that the coefficients of the Jones polynomial can be computed by counting solutions of the KW equations on a four-dimensional half-space, with certain boundary conditions that depend on a knot. The boundary conditions are defined by a "Nahm pole" away from the knot with a further singularity along the knot. In a previous paper, we gave a precise formulation of the Nahm pole boundary condition in the absence of knots, in the present paper, we do this in the more general case with knots included. We show that the KW equations with generalized Nahm pole boundary conditions are e…
In the present paper, we establish a gluing construction for the Nahm pole solutions to the Kapustin-Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing the two solutions together along the cylindrical end. In addition, we establish a local Kuranishi model for this gluing picture. As an application, we show that over any compact four-manifold with $S^3$ or $T^3$ boundary, there exists a Nahm pole solution to the obstruction pertur…
We develop a Kobayashi-Hitchin correspondence for the extended Bogomolny equations, i.e., the dimensionally reduced Kapustin-Witten equations, on the product of a compact Riemann surface $Σ$ with ${\mathbb R}^+_y$, with generalized Nahm pole boundary conditions at $y=0$. The correspondence is between solutions of these equations satisfying these singular boundary conditions and also limiting to flat connections as $y \to \infty$, and certain holomorphic data consisting of effective triplets $(\mathcal E, φ, L)$ where $(\mathcal E, φ)$ is a stable $\mathrm{SL}(n+1,\mathbb C)$ Higgs pair and $L …
For a 3-manifold $X$ and compact simple Lie group $G$, we study the expansions of polyhomogeneous Nahm pole solutions to the Kapustin-Witten equations over $X\times (0,+\infty)$. Let $y$ be the coordinate of $(0,+\infty)$, we prove that the sub-leading terms of a polyhomogeneous Nahm pole solution is smooth to the boundary when $y\to 0$ if and only if $X$ is an Einstein 3-manifold.
In this paper we develop a Kobayashi-Hitchin type correspondence between solutions of the extended Bogomolny equations on $Σ\times \RP$ with Nahm pole singularity at $Σ\times \{0\}$ and the Hitchin component of the stable $SL(2,\mathbb{R})$ Higgs bundle; this verifies a conjecture of Gaiotto and Witten. We also develop a partial Kobayashi-Hitchin correspondence for solutions with a knot singularity in this program, corresponding to the non-Hitchin components in the moduli space of stable $SL(2,\mathbb{R})$ Higgs bundles. We also prove existence and uniqueness of solutions with knot singulariti…
In this paper, we complete the proof of an equivalence given by Nye and Singer of the equivalence between calorons (instantons on $S^1\times R^3$) and solutions to Nahm's equations over the circle, both satisfying appropriate boundary conditions. Many of the key ingredients are provided by a third way of encoding the same data which involves twistors and complex geometry.
We consider solutions of Kapustin-Witten equation with Nahm pole boundary on $S^3\times \mathbb{R}^+$. These solutions are usually called Nahm pole solutions. In this paper, we will prove that there exists a constant $C>0$ such that $\|F_A\|_{L^2}\leq C$ for any Nahm pole solution $(A,ϕ)$.
Nahm's equations are viewed in a more general context where they appear as a vector field on a moduli space of co-Higgs bundles on the projective line. Zeros of this vector field correspond to torsion-free sheaves on a singular spectral curve which we translate in terms of a smooth curve in three-dimensional projective space. We also show how generalizations of Nahm's equations are required when the spectral curve is non-reduced and deduce the existence of non-classical conserved quantities in this situation.
Nahm's Equations and Rational Maps from $\mathbb{C}\mathrm{P}^1$ to $\mathbb{C}\mathrm{P}^n$
本文研究带一阶端点极点的 Nahm 方程，通过边界条件将其扩展模空间等同于从 $\mathbb{C}\mathrm{P}^1$ 到 $\mathbb{C}\mathrm{P}^n$ 的有理映射空间，并建立与完全旗流形有理映射模空间的辛同胚对应。
The extended Bogomolny equations on $\mathbb {R}^2 \times \mathbb {R}^+$ I: Nilpotent Higgs field
本研究针对$\u211d^2 \times \u211d^+$上带特定边界条件与幂零Higgs场的$SU(2)$规范扩展Bogomolny方程，借助Kobayashi-Hitchin对应关联全纯数据，实现解的完全分类。
The extended Bogomolny equations on $R^2 \times R^+$ with real symmetry breaking
本文构造了$R^2 \times R^+$上满足广义Nahm极边界条件与实对称性破缺渐近条件的扩展Bogomolny方程的解，该解是Taubes与Dimakis所构造瞬子解的类似物。
In this paper we establish a connection between free boundary minimal surfaces in a ball in $\mathbb{R}^3$ and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.
In this paper, we consider a free boundary problem with volume constraint. We show that positive minimizer is locally Lipschitz and the free boundary is analytic away from a singular set with Hausdorff dimension at most $n-8$.
Some Foundational Results for Free Boundary Brakke Flows
本文为自由边界Brakke流建立了几何与分析基础，引入单位正则和循环自由边界流，证明支撑集的回避原理，并引入内外流以证明匹配自由边界流的存在性。
Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary
本研究通过分析带严格凸边界闭曲面上的自由边界曲线缩短流的非坍缩性与收敛性，证明了带严格凸边界的黎曼2-圆盘上存在两条自由边界嵌入测地线及特定Morse指数的简单闭测地线，解决了Grayson定理的自由边界对应问题。
We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.
A Minkowski type inequality with free boundary in space forms
本文将空间形式中支撑在脐超曲面上的闵可夫斯基不等式推广到自由边界情形，建立了最优加权闵可夫斯基不等式的自由边界版本。
We prove an Allard-type regularity theorem for free-boundary minimal surfaces in Lipschitz domains locally modelled on convex polyhedra. We show that if such a minimal surface is sufficiently close to an appropriate free-boundary plane, then the surface is $C^{1,α}$ graphical over this plane. We apply our theorem to prove partial regularity results for free-boundary minimizing hypersurfaces, and relative isoperimetric regions.
We study inverse mean curvature flow with free boundary supported on geodesic spheres in hyperbolic space. Starting from any convex hypersurface inside a geodesic ball with a free boundary, the flow converges to a totally geodesic disk in finite time. Using the convergence result, we show a Willmore type inequality.
We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature $-$ including strictly convex domains of the Euclidean space $\mathbb{R}^3$.
In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem. We review several different techniques of constructing examples of embedded free boundary minimal surfaces in the unit ball. Next, we discuss some uniqueness results for free boundary minimal disks and the conjecture about the uniqueness of critical catenoid. We also discuss several Morse index estimates for free boundary min…
From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension $8$, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin's theorem is optimal.
A Family of Instanton-Invariants for Four-Manifolds and Their Relation to Khovanov Homology
本文基于物理论证，提出四流形上的一个参数族Haydys-Witten瞬子Floer同调群，通过维度约简和边界条件，建立与Khovanov同调的关系。
In this paper we investigate free boundary minimal surfaces in the unit ball in Euclidean 3-space, and by using holomorphic techniques we prove that intersection curves of free boundary minimal surfaces with the unit sphere are all circles.
We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula valid at all points, and use this to adapt the local regularity theorem of White to the free-boundary setting. We use Ilmanen's elliptic regularization procedure to prove existence of free-boundary Brakke flows.
We employ min-max techniques to show that the unit ball in $\mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
In the Euclidean unit three-ball, we construct compact, embedded, two-sided free boundary minimal surfaces with connected boundary and prescribed high genus, by a gluing construction tripling the equatorial disc. Aside from the equatorial disc itself, these are the first examples in the three-ball of compact free boundary minimal surfaces with connected boundary.
In this note we investigate free boundary minimal surfaces in the Euclidean 3-space, and by using holomorphic techniques developed by Fraser and Schoen we prove that the free boundary minimal annulus is the critical catenoid.
Using equivariant differential geometry, we provide a family of free boundary minimal surfaces in the unit ball.
Equivariant free boundary minimal discs and annuli in ellipsoids
本文利用等变变分方法在椭球中构造非平面自由边界极小圆盘，并证明每个椭球至少包含三个具有二面体对称性的嵌入自由边界极小圆环。
Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains
本研究针对一般有界凸区域上的自由边界凸古老曲线缩短流开展分类工作，完成了对应解的分类，为曲线缩短流领域的古老解研究提供了相关结果。
In this paper, we prove a family of sharp geometric inequalities for free boundary hypersurfaces in a ball in space forms.
Classification of convex ancient free boundary curve shortening flows in the disc
本文分类了圆盘内边界在圆上的凸古曲线缩短流，解决了该几何流的全局分类问题。
In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.
In this work, we investigate the existence of compact free-boundary minimal hypersurfaces immersed in several domains. Using an original integral identity for compact free-boundary minimal hypersurfaces that are immersed in a domain whose boundary is a regular level set, we study the case where this domain is a quadric or, more generally, a rotational domain. This existence study is done without topological restrictions. We also obtain a new gap theorem for free boundary hypersurfaces immersed in an Euclidean ball and in a rotational ellipsoid.
Free boundary saddle disks in the unit ball
本文在三维单位球中构造了无限族非平面、高斯曲率非正的自由边界鞍形圆盘，解决了相关存在性问题。
We first consider a uniqueness problem for embedded free boundary minimal annuli in the three-dimensional Euclidean unit half-ball. Then, we obtain symmetry properties for compact embedded free boundary minimal surfaces in the unit ball. Finally, we obtain several uniqueness results for the critical catenoid under symmetry conditions on the boundary. For example, we show that if an embedded free boundary minimal annulus whose boundary consists of two congruent components and has a reflection symmetry by a plane, then it is congruent to the critical catenoid.
In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 1$ and $m\geq 1$ ideal boundary components. 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, and possibly finitely many branched minimal spheres and free boundary minimal disks.
Free Boundary Stable Minimal Hypersurfaces in Positively Curved 4-Manifolds
该研究证明在正曲率4流形中，自由边界稳定极小超曲面的刚性由非负2中间Ricci曲率和标量曲率严格正性共同决定。
Ancient solutions to free boundary mean curvature flow
本文研究了自由边界平均曲率流中古老解的刚性性质，证明了极小超曲面与莫尔斯指数之间的关系，并构建了描述平均凸古老解的几何结构。
We study the blow-up analysis and qualitative behavior for a sequence of harmonic maps with free boundary from degenerating bordered Riemann surfaces with uniformly bounded energy. With the help of Pohozaev type constants associated to harmonic maps defined on degenerating collars, including vertical boundary collars and horizontal boundary collars, we establish a generalized energy identity.
Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to achieve a strong "Colding-Minicozzi" type min-max approximation result.
Using the linearized doubling methodology we carry out a PDE gluing construction of a discrete family of non-rotational properly immersed free boundary minimal annuli in the Euclidean unit 3-ball. The surfaces we construct resemble equatorial disks joined by half-catenoidal bridges at the boundary.
We study the free boundary Euler equations in two spatial dimensions. We prove that if the boundary is sufficiently regular, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity.
In the present paper, we study sharp isoperimetric inequalities for the first Steklov eigenvalue $σ_1$ on surfaces with fixed genus and large number $k$ of boundary components. We show that as $k\to \infty$ the free boundary minimal surfaces in the unit ball arising from the maximization of $σ_1$ converge to a closed minimal surface in the boundary sphere arising from the maximization of the first Laplace eigenvalue on the corresponding closed surface. For some genera, we prove that the corresponding areas converge at the optimal rate $\frac{\log k}{k}$. This result appears to provide the firs…
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