指定 Ricci 曲率泛函的 Palais-Smale 序列

Global hyperbolicity and Palais-Smale condition for action functionals in stationary spacetimes

In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients of the metric, etc. We prove that these technical assumptions admit a natural interpretation for the conformal structure (causality) of the manifold. As a consequence, any stationary spacetime with a complete timelike Killing vector field and a complete Cauchy hypersurface (thus, globally hyperbolic), is proved to be geodesically connected.

Fréchet 空间的全局微分同胚定理

A Global Diffeomorphism Theorem for Fréchet spaces

本文研究了 Fréchet 空间中局部微分同胚成为全局微分同胚的充分条件，通过将 Clarke 广义梯度理论推广至该空间并证明山路定理，得出了全局微分同胚定理。

Tangential category and critical point theory

Classical Ljusternik-Schnirelmann category is upper bounded by the number of critical points of any bounded from below differentiable functions of Palais-Smale type. Here we achieve an adaptation of this result for the tangential category of foliations. We introduce a weaker type of Palais-Smale function, obtaining a slight improvement in the classical theorem of critical points.

几何结理论中Palais-Smale条件的探讨

On the Palais-Smale condition in geometric knot theory

本文研究了几何结理论中多种能量函数在弧长参数化结子流形上满足Palais-Smale条件，并证明了这些能量存在最小化结及临界结的光滑性。

一种针对依赖特征值的泛函的变分方法

A variational method for functionals depending on eigenvalues

该研究提出了一套针对依赖黎曼流形特征值泛函的系统变分方法，通过将极小极大方法推广到局部Lipschitz泛函构造Palais-Smale序列，并证明了二维下两类特征值组合对应序列的收敛性。

具有指定曲率的平面闭曲线

Planar closed curves with prescribed curvature

本文利用变分方法并结合Struwe的单调性技巧，通过引入参数获得有界Palais-Smale序列，从而证明某些曲率函数类下具有指定曲率的平面闭曲线的存在性。

Compactness Theorems for Invariant Connections

Necessary and sufficient conditions are given for the Palais-Smale Condition C to hold for the Yang-Mills functional for connections that are invariant under a Lie group action on the manifold with orbits of codimension less than or equal to three. As an application the mountain pass lemma is used to give a simple proof of Wang's theorem on the existence of irreducible non-(anti-)selfdual instantons on S^2 x S^2 .

Smale猜想与极小极大理论

The Smale conjecture and min-max theory

本文利用极小曲面与极小极大理论，为$\mathbb{RP}^3$及所有透镜空间的Smale猜想提供了一个新证明，此前$\mathbb{RP}^3$情形由Bamler-Kleiner于2019年用Ricci流首次证明。

On the Existence of Critical Points to the Seiberg-Witten Functional

Considering that the Seiberg-Witten functional satisfies the Palais-Smale Condition, up to gauge equivalence, the Minimax Principle can be applied on the moduli space to prove the existence of critical points, which correspond to solutions of the second-order SW-equations, up to gauge equivalence.

Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds, I

We study Yamabe metrics, and the moduli space of Yamabe metrics, on an arbitrary closed 3-manifold M. The main focus is on the boundary behavior of the moduli space, i.e. the behavior of degenerating sequences of unit volume Yamabe metrics on M. It is proved that such degenerations, when non-trivial in a certain sense, are described by solutions of the static vacuum Einstein equations. Natural conditions are given for the non-triviality of degenerating sequences and relations with Palais-Smale sequences for the Einstein-Hilbert functional are explored. A number of new examples, both trivial an…

Ricci flow and contractibility of spaces of metrics

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group. This proves the Generalized Smale Conjecture. Our argument is independent of Hatcher's theorem in the $S^3$ case and in particular it gives a new proof of the $S^3$ case.

Two-dimensional curvature functionals with superquadratic growth

For two-dimensional, immersed closed surfaces $f:Σ\to \R^n$, we study the curvature functionals $\mathcal{E}^p(f)$ and $\mathcal{W}^p(f)$ with integrands $(1+|A|^2)^{p/2}$ and $(1+|H|^2)^{p/2}$, respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p > 2$. Our main result asserts that $W^{2,p}$ critical points are smooth in both cases. We also prove a compactness theorem for $\mathcal{W}^p$-bounded sequences. In the case of $\mathcal{E}^p$ this is just Langer's theorem \cite{langer85}, while for $\mathcal{W}^p$ we have to impose a bound for the Willmor…

Ricci flow and diffeomorphism groups of 3-manifolds

We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabai's theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than $S^3$ and $RP^3$ and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold $X$, the inclusion $\text{Isom} (X,g)\to \text{Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature.