Manuel Ritoré
Let $M$ be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of $M$ is a continuous and non-decreasing function. Particular cases are Hadamard manifolds and complete non-compact manifolds with strictly positive sectional curvatures.
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The Cartan-Hadamard isoperimetric inequality under controlled variation of curvature
本文在截面曲率受控变化条件下证明了任意维度的Cartan-Hadamard猜想，通过控制商极小化子的凸包并比较度量畸变与双曲等周轮廓盈余，给出关键证明步骤。
Nonexistence of isoperimetric sets in spaces of positive curvature
该研究构造了正曲率黎曼流形及凸集中小体积下等周集不存在的例子，证明维度下界3为最优，并首次给出无等周集的非塌缩非负曲率空间。
For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers. Then under an extra hypothesis on the geometry of $M$, we apply this result to prove some differentiability property of $I_M$ and a differential inequality satisfied by $I_M$, extending in this way well known results for compact manifolds, to this class of noncompact compl…
For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is a locally $\left(1-\frac{1}{n}\right)$-Hölder continuous function and so in particular it is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers. We prove also the equivalence of the weak and strong formulation of the isoperimetric profile function in complete Riemannian manifolds which is based on a lemma having its own int…
In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of t…
We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hoelder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs.
In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below assuming Gromov--Hausdorff asymptoticity to the suitable simply connected model of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by…
The isoperimetric inequality and CMC hypersurfaces in Cartan-Hadamard manifolds
本文通过常平均曲率超曲面的尖锐不等式及边界点对积分和Jacobi场估计，证明了$3\leq n\leq9$维Cartan-Hadamard流形中的欧几里得等周不等式，确立了相应维度的Cartan-Hadamard猜想。
On clusters and the multi-isoperimetric profile in Riemannian manifolds with bounded geometry
本文在具有有界几何的完备黎曼流形中证明等周簇的存在性、紧性及有界性，引入并证明多重等周剖面的Hölder连续性，从而将经典等周问题推广到黎曼等周簇背景。
Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds
本文针对具有下Ricci曲率界的非光滑RCD空间，通过原创变分估计与渐近质量分解方法，证明了尖锐等周比较定理与等周轮廓的维数凹性，推广了紧流形及欧氏凸体上的经典结果。
Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature
研究非负曲率有限维Alexandrov空间的等周剖面、等周集存在性与无穷远渐近结构的关系，证明线性体积增长等条件的等价性，并给出2维情形下等周集的存在性与拓扑刻画。
The Cartan-Hadamard conjecture in dimension five
本文证明了五维完备单连通非正截面曲率流形上的欧几里得等周不等式，确立了Cartan-Hadamard猜想，方法基于边界点对积分与Jacobi场估计，并推广至三维。
Pushforward of currents under Sobolev maps
该研究证明Sobolev映射可将模意义下几乎所有紧支撑整流推前为度量空间中的Ambrosio–Kirchheim整流，推导了相应的相对等周不等式，并正面解答了拟正则曲线最优Hölder连续性的公开问题。
For a complete noncompact connected Riemannian manifold with bounded geometry, we prove a compactness result for sequences of finite perimeter sets with uniformly bounded volume and perimeter in a larger space obtained by adding limit manifolds at infinity. We extends previous results contained in [Nar14a], in such a way that the generalized existence theorem, Theorem 1 of [Nar14a] is actually a generalized compactness theorem. The suitable modifications to the arguments and statements of the results in [Nar14a] are non-trivial. As a consequence we give a multipointed version of Theorem 1.1 of…
Isoperimetric minimizing movements and AC curves in spaces of measures
在具有 $L^p$ 密度和有限 $q$ 阶矩的概率测度空间 $\text{PL}_q^p(\mathbb{R}^n)$ 上定义完备度量结构 $\mathfrak{d}_q^p$，通过连续性方程的弱解（速度场满足 Sobolev 型条件）建立等周比的广义极小化移动的存在性并刻画 $\mathfrak{d}_q^p$-绝对连续曲线，同时刻画 $\infty$-Wasserstein 空间中的绝对连续曲线并证明 $W_\infty$ 的 Benamou-Brenier 公式。
Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from $\mathrm{RCD}(K,N)$ spaces to $\mathrm{CAT}(0)$ spaces
本文研究RCD(K,N)空间到CAT(0)空间调和映射的正则性，利用最优输运与热流相互作用及粘性摄动论证，证明其Lipschitz连续性并建立含Hessian项的Bochner-Eells-Sampson不等式，推广了经典理论。
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