Camillo De Lellis, Emanuele Spadaro
This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multiple valued functions.
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Geometric regularity of blow-up limits of the Kähler-Ricci flow
本文证明了 Kähler-Ricci 流 Type I 爆破极限的几何正则性，得到极限流在 Gromov-Hausdorff 和 Gromov-$W_1$ 距离下时间连续，且奇异集余维数至少为 4。
We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; this partial regularity is such that we can conclude the tangent cone is unique. The proof follows closely the boundary regularity result given by Hardt and Simon in [9].
We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; this partial regularity is such that we can conclude the tangent cone is unique. The proof closely follows that giving the boundary regularity result of Hardt and Simon.
Area-minimizing horizontal graphs with low-regularity in the sub-Finsler Heisenberg group $\mathbb{H}^1$
研究次芬斯勒海森堡群$\mathbb{H}^1$中低正则性面积极小水平图，构造以水平线或共点水平半直线有限并为奇异集的欧氏局部利普希茨整面积极小水平图，还给出多类面积极小锥例子。
In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area minimizing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the latter. This note establishes a new a priori estimate on the excess measure of an area minimizing current, together with several statements concerning approximations with Lipschitz multiple valued graphs. Our new a priori estimate is an higher integrability type result, which has a counterpart in the theory of Dir-minimizing multiple valued functions and plays …
We adapt to an infinite dimensional ambient space E.R. Reifenberg's epiperimetric inequality and a quantitative version of D. Preiss' second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable $G$ chain in $\ell_2$ is dense in its support, whenever the group $G$ of coefficients is so that $\{\|g\| : g \in G \}$ is discrete and closed.
On the fundamental regularity theorem for mass-minimizing flat chains
本文在赋范阿贝尔群系数的平坦链理论中，给出了群元素满足质量最小化链基本正则性原理的充要条件，刻画了弱接近重数 g 圆盘时在更小球内成为 C^{1,α} 扰动的判别准则。
This lecture notes are an expanded version of the course given at the ERC-School on Geometric Measure Theory and Real Analysis, held in Pisa, September 30th - October 30th 2013. The lectures aim to explain the main steps of a new proof of the partial regularity of area minimizing integer rectifiable currents in higher codimension, due originally to F. Almgren, which is contained in a series of papers in collaboration with C. De Lellis (University of Zurich).
This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely the construction of a center manifold, i.e. an approximate average of the sheets of an almost flat area minimizing current. Such center manifold is complemented with a Lipschitz multi-valued map on its normal bundle, which approximates the current with a highe degree of accuracy. In the third and final paper these objects are used to conclude a new proof of Almgren's celebrated dimension bound …
We analyze the asymptotic behavior of a $2$-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
Existence of flat blowups at boundary points of anisotropic minimizing hypercurrents
本文证明，在空间变量连续、法向变量$C^{2,{\rm Dini}}$的各向异性能量下，几乎极小的$m$维流若边界可微，则每个边界点存在平坦爆破。
We show $C^{1,α}$-regularity for energy minimizing maps from a 2-dimensional Riemannian manifold into a Finsler space $(\R^n, F)$ with a Finsler structure $F(u,X)$.
Generic regularity for minimizing hypersurfaces in dimensions 9 and 10
研究9和10维中最小化超曲面的奇异性扰动问题，通过正则性理论证明奇异点可被扰动消除。
In this article, we show that, for any compact 3-manifold, there is a $C^{1}$ volume-minimizing one-dimensional foliation. More generally, we show the existence of mass-minimizing rectifiable sections of sphere bundles without isolated "pole points" in the base manifold. This same analysis is used to show that the examples, due to Sharon Pedersen, of potentially volume-minimizing rectifiable sections (rectifiable foliations) of the unit tangent bundle to $S^{2n+1}$ are not, in fact, volume minimizing.
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。
We give new examples of entire area-minimizing t-graphs in the subriemannian Heisenberg group H^1. Most of the examples are locally lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a horizontal line or a finite number of horizontal halflines extending from a given point. Amongst them, a large family of area-minimizing cones is obtained.
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$n-10-\epsilon_n$。
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…
Interior regularity of area minimizing currents within a $C^{2,α}$-submanifold
本文证明了在 $C^{2,α}$ 类嵌入子流形内，面积极小整流的内部奇异集的 Hausdorff 维数不超过 $m-2$，确立了任意余维数下该流的内部正则性理论。
In this article we extend to generic $p$-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $p=2$. We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of the map around them, as done by Cheeger and Naber in 2013. Then, adapting the work of Naber and Valtorta, we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is $k$…
Improved regularity for minimizing capillary hypersurfaces
本文研究了极小毛细超曲面的奇点集正则性，证明其奇点集余维数至少为4，并在毛细角接近0、π/2或π时基于与单相Bernoulli问题的联系给出改进估计。
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。
In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent $L^p$ bounds for $\nabla^k f$ that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in $W^{4,p}$ for all $1\le p<5/4$. Further, for minimizing biharmonic maps from $Ω\subset \mathbb{R}^5$, we determine a uniform bound on the number of singular points in a compact set. Finally, using dimension reduction arguments, we extend these results to minimizing and stationary biharmonic maps …
We prove regularity estimates in weighted Sobolev spaces for the $L^2$-eigenfunctions of Schrödinger type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual $N$-body Hamiltonians with Coulomb-type singular potentials are covered by our result: in that case, the weight is $δ_{\mathcal{F}}(x) := \min \{ d(x, \bigcup \mathcal{F}), 1\}$, where $d(x, \bigcup \mathcal{F})$ is the usual euclidean distance to the union $\bigcup\mathcal{F}$ of the set of collision planes $\bigcup\mathcal{F}$. The proof is based on blow-ups of man…
Regularity of minimizing $p$-harmonic maps from $B^3$ to $\mathbb{S}^3$
本文证明从三维球到三维球面的极小p-调和映射在p>1.44时具有局部C^{1,α}正则性，填补了先前正则性区间的空白，并覆盖所有p≥2及次二次范围。
Energy minimizing harmonic maps between manifolds are known to be smooth outside a rectifiable set of codimension $3$, called the singular set. The possibility that this set is not a manifold, but has arbitrarily many small gaps in it, is not excluded in general. Here we prove that some part of the singular set - characterized by topological and analytic properties of tangent maps - is a topological manifold. In the special case of maps into the sphere ${\mathbb S}^2$, we conclude that the whole top-dimensional part of the singular set is a manifold - this generalizes a similar result in two-d…
In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic analytic distribution of rank 2 on a 3-dimensional analytic manifold, we investigate the size of the set of points that can be reached by singular horizontal paths starting from a given point and prove that it has Hausdorff dimension at most 1. In fact, provided that the lengths of the singular curves under consideration are bounded with respect to a given complete Riemannian metric, we demonstrate that such a set is a semianalytic cu…
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 …
Let B be a fiber bundle with compact fiber F over a compact Riemannian n-manifold M. There is a natural Riemannian metric on the total space B consistent with the metric on M. With respect to that metric, the volume of a rectifiable section s:M--> B is the mass of the image s(M) as a rectifiable n-current in B. Theorem: For any homology class of sections of B, there is a mass-minimizing Cartesian current T representing that homology class which is the graph of a C^1 section on an open dense subset of M.
We construct a branched center manifold in a neighborhood of a singular point of a $2$-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensional integral currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
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。
Regularity of minimizing $p$-harmonic maps into spheres and sharp Kato inequality
该研究针对取值于3维球面的3维单位球上的极小$p$-调和映射，结合经典方法与新工具拓展了已知正则性的$p$值区间，还证明了二维区域下的精确Kato不等式。
Existence and partial regularity of Legendrian area-minimizing currents
本文证明接触流形中局部质量极小的Legendrian积分流具有开且稠密的正则集，并据此建立Heisenberg群中Legendrian Plateau问题及闭接触流形同调类中极小流的存在性与部分正则性。
Optimal higher regularity for biharmonic maps via quantitative stratification
本文利用定量分层方法改进Breiner-Lamm关于双调和映射的几乎最优正则性结果，并得到极小化双调和映射的最优正则性。
Regularity for free boundary surfaces minimizing degenerate area functionals
本文针对退化加权面积泛函的近似极小值点，建立了自由边界的ε-正则性定理，证明了边界为正则曲面且正交接触边界，并给出了奇异集的Hausdorff维数上界。
This a survey on a series of recent papers in collaboration with Emanuele Spadaro on the regularity of area-minimizing currents in codimension higher than $1$.
Quantitative stratification and optimal regularity for harmonic almost complex structures
本文通过定量分层方法改进了谐波近复结构的正则性理论，建立了最优正则性结果。
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…
We study the existence and regularity of energy-minimizing harmonic almost complex structures. We have proved results similar to the theory of harmonic maps, notably the classical results of Schoen-Uhlenbeck and recent advance by Cheeger-Naber.
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。
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