Camillo De Lellis, Guido De Philippis, Jonas Hirsch
We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $Γ\subset \mathbb R^4$ such that the unique area minimizing surface spanned by $Γ$ has infinite topology. Furthermore the metric is almost Kähler and the area minimizing surface is calibrated.
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We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.
Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in $\mathbb R^3$
该研究证明了三维空间中带预定接触角液滴的最小化运动解与光滑平均曲率流等价，为两类平均曲率运动描述建立了一致性。
We show that any minimizing hypercone can be perturbed into one side to a properly embedded smooth minimizing hypersurface in the Euclidean space, and every viscosity mean convex cone admits a properly embedded smooth mean convex self-expander asymptotic to it near infinity. These two together confirm a conjecture of Lawson \cite[Problem 5.7]{Brothers86_OpenProblems}.
Area-minimizing submanifolds are not generically smooth
本文证明面积最小化子流形并非一般光滑，解决White猜想，并给出其奇异集Hausdorff维数的下界$\max\{d-5,d-c\}$。
We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (e.g. those of Reifenberg, Taylor and White), we need no a priori assumptions on the structure of the cone (e.g. integrability). Moreover, if the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new $\varepsilon$-regularity result for almost …
In this paper we show that every area minimizing cone C^{n-1} in R^n can be approximated by entirely smooth area minimizing hypersurfaces. This extensively uses hyperbolic unfoldings of such hypersurfaces and the resulting potential theory for the Jacobi field operator. Applications include the splitting theorem in scalar curvature geometry saying that any (n-1)-dim. homology class of a compact manifold M^n with positive scalar curvature can be represented by a smooth (!) compact hypersurface that admits a positive scalar curvature metric.
Minimizing movements for forced anisotropic curvature flow of droplets
本文研究非均匀水平超平面上液滴的受迫各向异性曲率流，建立了光滑流和1/2-赫尔德连续极小化移动解的存在性，并探讨了其比较原理、一致有界性及与光滑流的一致性。
Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces
研究模2同调中面积最小化子流形的光滑性，证明其一般非光滑（除特定情况），解决怀特猜想，还建立奇异集豪斯多夫维数下界，并证明$\rpt$维罗内塞极小嵌入上的锥是模2面积最小化的。
Smooth maps minimizing the energy and the calibrated geometry
该研究将校准子流形概念推广至光滑映射，验证了微分几何中相关实例的适配性，同时给出黎曼流形同伦类光滑映射能量下界，探究特殊和乐群流形上恒等映射极小化的能量泛函。
Improved generic regularity of codimension-1 minimizing integral currents
本文证明了对任意光滑闭定向余维1子流形，存在任意小扰动使其极小积分流形在余维至少9+ε的奇异集外光滑，改进了先前在9、10维的结果，核心是新的奇异集估计与超线性衰减方法。
By Federer and Fleming there exist at least one mass-minimizing normal current in every real-valued homology class of a Riemannian manifold. However the regularity of the mass-minimizing currents and their distributions may generally be quite complicated. In this paper we shall study how to construct nice metrics so that (as functionals over smooth forms) almost all homologically mass-minimizing currents are just linear combinations over smooth submanifolds.
Not all sub-Riemannian minimizing geodesics are smooth
本文否定了次黎曼长度极小化子必然光滑的猜想，通过构造一个实解析（甚至多项式）次黎曼结构，证明了存在属于 $C^2$ 但非 $C^3$ 的长度极小化子。
We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold $(M,h)$ without boundary. First, under the assumption that $(M,h)$ is the euclidean 3-space endowed with a semi-perturbed metric with perturbation small in $C^1$ norm and of compact support, we prove that if there is some point $\bar{x} \in M$ with scalar curvature $R^M(\bar{x})>0$ then there exists a smooth embedding $f:S^2 \hookrightarrow M$ minimizing the Willmore functional $1/4\int |H|^2$, where $H$ is the mean curvature. Second, assuming that $(M,h)$ is of bounded geometry (i.e. bo…
Allard-type regularity theory for area minimizing currents at boundaries with arbitrary multiplicity
本文推广了Allard边界正则性定理，研究具有光滑边界且边界取任意重数的面积最小化流，并由此得出重要结构性结论。
Generic regularity for minimizing hypersurfaces in dimension 11
本文研究面积极小化超曲面的一般正则性，证明11维情形下一般光滑，并在更高维中通过小扰动将奇异集维数控制为至多$n-10-\epsilon_n$。
An essential one sided boundary singularity for a $3$-dimensional area minimizing current in $\mathbb{R}^5$
本文构造了 $\mathbb{R}^5$ 中具有平坦切锥的本质单侧边界奇点的 $3$ 维面积极小流，证明了高边界重数下的边界正则性理论在维度上是精确的，并发展了相关边界正则性理论。
We construct a sequence of smooth minimizing surfaces in a sequence of metrics converging to the standard Euclidean metric, so that they have diverging $L^2$ norm of second fundamental form.
Minkowski content estimates for generic area minimizing hypersurfaces
该研究通过扰动方法证明了面积最小超曲面的奇异性集的Minkowski维数低于n-9。
The genus-minimizing property of algebraic curves
一个可行且尚未证明的猜想是，如果$X$是一个光滑的代数曲面，而$C$是$X$中的一条光滑的代数曲线，则$C$在其同调类中实现了所有光滑嵌入的$2$ -流形中的最小可能亏格。本文对一大类曲面$X$，在$C$的法丛具有正度的假设下，给出了这个猜想的一个证明.
Minimizing tangent maps into complex projective space
该研究分类了从二维球面$\text{S}^2$、三维球面$\text{S}^3$到复射影空间$\text{CP}^n$的非平凡光滑调和映射，明确其极小映射的具体形式
Suppose that $N$ is a smooth manifold with a smooth Riemannian metric $g_0$, and that $Γ$ is a smooth submanifold of $N$. This paper proves that for a generic (in the sense of Baire category) smooth metric $g$ conformal to $g_0$, if $F$ is any simple $g$-minimal immersion of a closed manifold into N, then $F$ is transverse to $Γ$ and $F$ is self-transverse. The theorem remains true with "transverse" and "self-transverse" replaced by "strongly transverse" and "strongly self-transverse". The theorem also holds for hypersurfaces of constant mean curvature or, more generally, of prescribed mean cu…
Every finite graph arises as the singular set of a compact $3$-d calibrated area minimizing surface
本文基于未发表的想法，在具有非零第三Betti数的紧致光滑6-流形上构造校准的3维同调面积最小化曲面，使其奇异集精确实现任意给定的有限图嵌入。
We consider the class of all conformal mappings from a compact Riemann surface into the threedimensional or fourdimensional Euclidean space. A sequence in this class with bounded Willmore functional is shown to have a sequence of conformal transformations of the target space, such that a subsequence of the transformed sequence converges. This implies that there exists a smooth conformal mapping, which minimizes the Willmore functional in this class. For this purpose we extend the quaternionic function theory of Pedit and Pinkall to square integrable Hopf fields. In particular, we proof the Plu…
We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^1 neighborhood.
Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing" maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps -- but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in $M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a given homotopy class. When $(S,h)$ is fixed, smooth minimizing maps …
On Conformal Flexibility of Completeness and Minimizing Geodesics on Hilbert Manifolds
研究希尔伯特流形上完备性相关问题，通过共形变形得到度量和测地线完备且有长度极小化测地线的代表，同时建立了度量完备性的局部灵活性现象，即能在保持测地线完备性的同时破坏度量完备性。
On a conjecture of Almgren II: area-minimizing submanifolds with fractal singular sets on almost any manifold
本文证明在几乎所有光滑流形上都存在具有分形奇点集的面积最小子流形，解决了阿尔梅伦猜想的部分问题。
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。
Homologically area-minimizing surfaces that cannot be calibrated
该研究打破高余维下面积极小曲面可被校准的固有认知，证明无法校准的同调面积极小曲面对应的度量构成非空开集，还解答了领域内多个公开猜想。
Quantitative stability of the total $Q$-curvature near minimizing metrics
该研究在正性假设下证明了闭黎曼流形上$k$阶总$Q$-曲率泛函在极小化度量附近的定量估计，推广了$k=1$的已有结果，并构造了亏空控制距离更高次幂的退化例子。
Hardt-Simon proved that every area-minimizing hypercone $\mathbf{C}$ having only an isolated singularity fits into a foliation of $\mathbb{R}^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $\mathbf{C}$. In this paper we prove that if a stationary $n$-varifold $M$ in the unit ball $B_1 \subset \mathbb{R}^{n+1}$ lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone $\mathbf{C}^{3,3}$), then $\mathrm{spt} M \cap B_{1/2}$ is a $C^{1,α}$ perturbation of either the cone itself, or some leaf of its associated foliation. In particular, we show that singul…
The paper studies how to extend local calibration pairs to global ones in various situations. As a result, new discoveries involving mass-minimizing properties are exhibited. In particular, we show that a $\mathbb R$-homologically nontrivial connected submanifold $M$ of a smooth Riemannian manifold $X$ is homologically mass-minimizing for some metrics in the same conformal class. Moreover, several generalizations for $M$ with multiple connected components or for a mutually disjoint collection (see §3.5) are obtained. For a submanifold with certain singularities, we also establish an extension …
The fine structure of the singular set of area-minimizing integral currents II: rectifiability of flat singular points with singularity degree larger than $1$
本文研究高余维面积极小积分流的平坦奇异点，证明奇异度严格大于1的点集具有$(m-2)$-可求长性，为完整刻画奇异集结构奠定关键一步。
We introduce and study co-dimension one area-minimizing locally rectifiable currents $T$ with $C^{1,α}$ tangentially immersed boundary: $\partial T$ is locally a finite sum of orientable co-dimension two submanifolds which only intersect tangentially with equal orientation. We show that any such $T$ is supported in a smooth hypersurface near any point on the support of $\partial T$ where $T$ has tangent cone which is a hyperplane with constant orientation but non-constant multiplicity. We also introduce and study co-dimensional one area-minimizing locally rectifiable currents $T$ with boundary…
Area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below
本文利用Cheeger-Colding理论，研究Ricci曲率下有界流形中面积极小超曲面在Gromov-Hausdorff收敛下的体积连续性，并通过blowing up论证给出极限超曲面奇异集的精确维数估计。
We prove that in a class of non-equiregular sub-Riemannian manifolds corners are not length minimizing. This extends the results [4]. As an application of our main result we complete and simplify the analysis in [6], showing that in a 4-dimensional sub-Riemannian structure suggested by Agrachev and Gauthier all length-minimizing curves are smooth.
Let $α$ be a polygonal Jordan curve in $\bfR^3$. We show that if $α$ satisfies certain conditions, then the least-area Douglas-Radó disk in $\bfR^3$ with boundary $α$ is unique and is a smooth graph. As our conditions on $α$ are not included amongst previously known conditions for embeddedness, we are enlarging the set of Jordan curves in $\bfR^3$ which are known to be spanned by an embedded least-area disk. As an application, we consider the conjugate surface construction method for minimal surfaces. With our result we can apply this method to a wider range of complete catenoid-ended minimal …
Sharp regularity of sub-Riemannian length-minimizing curves
本文研究了亚里叶米安结构中长度最短曲线的正则性，证明了此类曲线至少为C²类，并指出[6]中的定理1.1是尖锐的。
In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and $C^{2,α}$-close to the given sub manifold. We show also a version with variable metric on the manifold. The techniques used are, among other, the standards outils of linear elliptic analysis and comparison theorems of riemannian geometry, Allard's regularity theorem for minimizing varifolds, the isometric immersion theorem of Nash and a paramet…
Interior singularity and branching of geodesics in real-analytic sub-Riemannian manifolds
研究实解析子黎曼流形中严格异常极小测地线的正则性和分支特性，通过构造例子解决长期未解问题，并在 Carnot 组中证明非光滑和分支极小测地线的存在。
Homologically area-minimizing surfaces mod $v$ have at worst codimension 2 singular sets asymptotically
该研究在De Lellis等人模v面积极小流形正则性结果基础上，证明固定同调类且v充分大时，模v面积极小流形为整电流，奇异集余维更高，拓展了相关正则性理论。
$\mathbb{Z}_k$-smooth approximation of mod $k$ integral cycles
We prove that every mod $k$ integral cycle $T$ of sufficiently high codimension in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by a cycle which is a $\mathbb{Z}_k$-smooth submanifold $Σ$ of nearly the same area, up to a singular set of codimension 5. If in addition $T$ belongs to a mod $k$ homology class admitting an embedded $\mathbb{Z}_k$-smooth manifold representative, then $Σ$ can be taken fully regular, up to the codimension one stratum of classical singularities. This extends the smooth approximation theory to cycles with coefficients in $\mathbb{Z}/k\m…
For any $\mathbb{Q}$-Gorenstein klt singularity $(X,o)$, we introduce a normalized volume function $\widehat{\rm vol}$ that is defined on the space of real valuations centered at $o$ and consider the problem of minimizing $\widehat{\rm vol}$. We prove that the normalized volume has a uniform positive lower bound by proving an Izumi type estimate for any $\mathbb{Q}$-Gorenstein klt singularity. Furthermore, by proving a properness estimate, we show that the set of real valuations with uniformly bounded normalized volumes is compact, and hence reduce the existence of minimizers for the normalize…
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