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

The Palais-Smale condition for contact type energy levels for convex lagrangian systems

We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Ma ne's critical value of the lift of the Lagrangian to the universal cover, almost all energy levels have conjugate points. We prove that if the energy level [E=k] is of contact type and M is not the 2-torus then the free time action functional of L+k satisfies the Palais-Smale condition.

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。

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。

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。

Variational Principles for immersed Surfaces with $L^2$-bounded Second Fundamental Form

In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension. We explain how the same approach can solve constraint minimization problems for the Willmore functional. We show in particular that, for a given closed surface and a given conformal class for this surface, there is an immersion in $R^m$, away possibly from isolated branched points, which minimizes the Willmore …

Existence of variationally defined curves in complete Riemannian manifolds

We present a method for proving the existence of solutions to a class of one dimensional variational problems. The method is demonstrated by two examples of optimal interpolation problems which are motivated by engineering applications. In each case we prove that the variational problem satisfies the Palais-Smale condition and the existence of a minimal solution and lower bounds for the number of stationary curves then follow.

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 .

Variational Aspects of the Seiberg-Witten Functional

The Seiberg-Witten equations that have recently found important applications for four-dimensional geometry are the Euler-Lagrange equations for a functional involving a connection $A$ on a line bundle $L$ and a section $ϕ$ of another bundle $W^+$ constructed from $L$ and a spinor bundle on a given four-dimensional Riemannian manifold. We show the regularity of weak solutions and the Palais-Smale condition for this functional.

A Compactness Theorem for Invariant Connections

Necessary and sufficient conditions are given for the Palais-Smale Condition C to hold for the Yang-Mills functional for invariant connections on a principal bundle over a compact manifold of any dimension. It is assumed that the connections are invariant under the action of a compact Lie group on the manifold, and that all orbits of the action have codimension three or less.

On the energy functional on Finsler manifolds and applications to stationary spacetimes

In this paper we first study some global properties of the energy functional on a non-reversible Finsler manifold. In particular we present a fully detailed proof of the Palais--Smale condition under the completeness of the Finsler metric. Moreover we define a Finsler metric of Randers type, which we call Fermat metric, associated to a conformally standard stationary spacetime. We shall study the influence of the Fermat metric on the causal properties of the spacetime, mainly the global hyperbolicity. Moreover we study the relations between the energy functional of the Fermat metric and the Fe…

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.

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…

全局分析：黎曼流形上的非线性分析及其变分方法导论

Global Analysis: An Introduction to Nonlinear Analysis and Its Variational Methods on Riemannian Manifolds

本书系统介绍黎曼流形上的全局分析，融合几何、分析与PDE，涵盖指标定理、几何流及变分方法，强调显式证明与几何假设。