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2026-03-26
Tian Lan, Dorian Martino, Tristan Rivière
We review recent progress concerning the analysis of Lagrangians on immersions into $\mathbb{R}^d$ depending on the first and second fundamental forms and their covariant derivatives.
2026-03-26
Jingyi Chen, Xiaoli Han, Jiayu Li, Jun Sun
We prove that the tangent cone at the first blow-up time of the mean curvature flow of a closed symplectic surface in a compact Kähler-Einstein surface consists of a finite union of planes in $\mathbf{R}^4$. Furthermore, when the flow develops a Type I* singularity at $(X_0,T)$, then the tangent cone is a holomorphic cone.
2025-12-26
Shiguang Ma, Jie Qing
In this paper, we give an exposition of our recent work on nonlinear potential theory in conformal geometry. We apply nonlinear potential theory to study $p$-Laplace equations arising from conformal geometry and, in particular, the problems related to the asymptotic behavior near and the size of singularities in conformal geometry.
2025-12-26
2025-12-26
Hosea Wondo, Zhou Zhang
One of the significant motivations for studying the Kähler-Ricci flow is its relation to the Analytic Minimal Model Program as initiated by Gang Tian. Carrying out this classification program requires careful analysis of the flow metric, particularly when it encounters singularities. In this note, we survey some results pertaining to the limit for the Kähler-Ricci flow.
2025-12-26
Jianchun Chu, Feng Wang
Degenerate complex Monge-Ampère equations arise naturally in the study of geometry of singular varieties. In this paper, we prove gradient estimate and $W^{3,p}$ estimate for a class of degenerate complex Monge-Ampère equations.
2025-09-15
Albo Carlos Cavalheiro
In this paper we are interested in the existence of solutions for Dirichlet problem associated to the degenerate nonlinear elliptic equations in the setting of the weighted Sobolev spaces $W^{1,p}_ 0 (Ω,ω).$
2025-09-15
Xizheng Zhang, Xiang Li, Dunyan Yan
In this paper, we will study the sharp estimates of $p$-adic weighted Hardy operator on central and noncentral weighted $p$-adic Morrey spaces. Moreover, we can obtain the sharp bound of generalized $p$-adic Hardy operator. In addition, the commutator that is generated by the generalized $p$-adic Hardy operator and the central BMO function is also bounded on $p$-adic Morrey spaces.
2025-09-15
Chao Wang
In this paper, under the asymptotically regular condition, we investigate the existence of a unique common fixed point for a pair of generalized nonlinear mappings in the framework of complete metric spaces. Additionally, we extend our analysis to the case involving two control functions. Our work generalizes some results in recent papers.
2025-06-25
Cuifang Si, Shicheng Xu
In this note, we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it is Einstein, i.e. ${\rm Ric}_g =λg$ for some real number $λ.$ (2) A compact Einstein manifold with a non-vanishing and almost maximal volume entropy is hyperbolic. (3) A compact Einstein manifold admitting a uniform local rewinding almost maximal volume is isometric to a space form.