Baris Coskunuzer
We show that for a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. We also show that the same is true for absolutely area minimizing surfaces.
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Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the boundary of M which are nullhomotopic in M. We also show that the set of all simple closed curves in the boundary of M which bound unique absolutely area minimizing surfaces in M is dense in the space of simple closed curves in the boundary of M which are nullhomologous in M.
The purpose of this short note is to prove the uniqueness of Hamiltonian volume minimizing Lagrangian submanifolds which are Hamiltonian isotopic to RP^n in CP^n modulo isometric group actions.
On the uniqueness of energy-minimizing curves in constrained spaces
该研究针对约束空间中固定端点的能量最小化曲线唯一性问题展开，证明固定一个端点时，使能量最小化曲线不唯一的另一端点集合无内点，推进了几何变分问题的相关理论。
We consider $2$-dimensional integer rectifiable currents which are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by a suitable modification of White's original theorem for area minimizing currents in the euclidean space. This note is also the first step in a regularity program for semicalibrated $2$-dimensional currents and spherical cross sections of $3$-dimensional area minimizing cones.
We prove that tangent cones at singular boundary points of a two-dimensional current almost area minimizing are unique. Following the ideas exposed by White in [8], the result is achieved by combining a suitable epiperimetric inequality and an almost-monotonicity formula for the mass at boundary points.
Almost minimizing Yang$-$Mills fields: log-epiperimetric inequality, non-concentration, and uniqueness of tangents
本文为任意维度Yang-Mills场建立直接的对数周长不等式，并据此证明能量极小化Yang-Mills场和$ω$-ASD连接的孤立奇点切锥唯一性，同时通过Luckhaus型引理排除blow-up序列的曲率集中。
Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball
该研究针对满足体积约束的局部能量最小化集合，通过证明庞加莱型不等式，分类了单位球、半空间及楔形域中的此类集合，并证实其相对边界具有光滑性。
The Fine Structure of the Singular Set of Area-Minimizing Integral Currents III: Frequency 1 Flat Singular Points and $\mathcal{H}^{m-2}$-a.e. Uniqueness of Tangent Cones
本文证明了光滑 Riemannian 流形中余维数大于1的面积最小化整流单体在 $\mathcal{H}^{m-2}$-几乎处处具有唯一的切锥，并结合先前工作得出其奇异集是可数 $(m-2)$-可求长的。
On the uniqueness of tangent cones to area minimizing currents at boundaries with arbitrary multiplicity
本文证明任意整数边界重数的面积最小化流在密度 $Q/2$ 边界点处切锥唯一且收敛具有幂次速率，并由此导出支撑高重数边界正则性理论的分解定理。
Uniqueness in the Plateau problem near Quadratic cones
本文研究单位球内边界为极小化二次锥链环小扰动的极小超曲面，证明其由边界条件唯一确定，从而得到该类边界条件下 Plateau 问题解的唯一性。
We show that the cylindrical tangent cone $C\times \mathbf{R}$ for an area-minimizing hypersurface is unique, where $C$ is the Simons cone $C_S= C(S^3\times S^3)$. Previously Simon proved a uniqueness result for cylindrical tangent cones that applies to a large class of cones $C$, however not to the Simons cone. The main new difficulty is that the cylindrical cone $C_S\times \mathbf{R}$ is not integrable, and we need to develop a suitable replacement for Simon's infinite dimensional Lojasiewicz inequality in the setting of tangent cones with non-isolated singularities.
In this paper we continue to study the connection among the area minimizing problem, certain area functional and the Dirichlet problem of minimal surface equations in a class of conformal cones with a similar motivation from \cite{GZ20}. These cones are certain generalizations of hyperbolic spaces. We describe the structure of area minimizing $n$-nteger multiplicity currents in bounded $C^2$ conformal cones with prescribed $C^1$ graphical boundary via a minimizing problem of these area functionals. As an application we solve the corresponding Dirichlet problem of minimal surface equations unde…
Uniqueness of Cylindrical Tangent Cones $C_{p,q} \times \mathbb{R}$
研究面积最小超曲面在 $\mathbb{R}^9$ 中柱形切锥 $C(\mathbb{S}^2 \times \mathbb{S}^4) \times \mathbb{R}$ 的唯一性，完善了 Simon 和 Székelyhidi 对高维切锥唯一性的证明。
Analysis of singularities of area-minimizing currents, Part II: a uniform height bound, estimates away from branch points of rapid decay, and uniqueness of tangent cones
本文为n维面积最小化可求长流T建立新框架，给出高度估计和衰减估计。结合第一部分定理，得出T在几乎每个\({\mathcal H}^{n - 2}\)点有唯一切锥，其奇点集可数\((n - 2)\)可求长，分支点一般快速衰减到唯一切平面。
Uniqueness in the Plateau problem for calibrated currents
该研究针对校准流的有向Plateau问题，结合面积最小化流的边界正则性理论与Frank Morgan风格的唯一延拓论证，证明了具连通$C^{3,α}$边界的紧支撑光滑校准积分流的解唯一性，还推广至连续校准积分平链，并在余维1情形给出以校准形式正则性表述的唯一性充分条件。
Area-minimizing capillary cones
本研究构造了具有双正交对称群的非平坦最小容量锥体，证明其在特定接触角下最小化，并推进了接触角接近 π/2 的容量正则性理论。
On the asymptotic properties of solutions to one-phase free boundary problems
本文研究单相Bernoulli问题解的结构，在特定条件下证明了爆破唯一性及无穷远处刚性结果，是首次针对该问题非极小化解在奇点处的爆破/爆破下唯一性研究。
We prove that a strictly stable minimal $C^2_h$ intrinsic graph G is locally area-minimizing, i.e. given any $C^1_h$ graph $S$ with the same boundary, $\text{Area}(G)<\text{Area}(S)$ unless $G=S$. As a consequence we show the existence and the uniqueness of $C^\infty$ minimal graphs with prescribed small boundary datum.
Consider an area minimizing current modulo $p$ of dimension $m$ in a smooth Riemannian manifold of dimension $m+1$. We prove that its interior singular set is, up to a relatively closed set of dimension at most $m-2$, a $C^{1,α}$ submanifold of dimension $m-1$ at which, locally, $N\leq p$ regular sheets of the current join transversally, each sheet counted with a positive multiplicity $k_i$ so that $\sum_i k_i = p$. This completes the analysis of the structure of the singular set of area minimizing hypersurfaces modulo $p$, initiated by J. Taylor for $m=2$ and $p=3$ and extended by the authors…
We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain natural geometric asymptotic constraints, in terms of certain associated varifolds which can be enumerated explicitely. One consequence of this result is the uniqueness of the area minimizing examples. Another is the asymptotic nonexistence of stable compact embedded minimal surfaces of positive genus bounded by two convex curves in parallel planes.
Area minimizing hypersurfaces modulo $p$: a geometric free-boundary problem
本文研究完备 Riemann 流形中模 $p$ 面积极小流的局部结构，证明其向具有脊的切锥唯一收敛并衰减，从而给出奇数模数及偶数模数相应条件下的自由边界分解。
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