arXiv research

We extend the Palais-Smale condition to Keller's $C_c^1$-functionals on Fréchet spaces. Using this condition together with Ekeland's variational principle, we obtain some results regarding the existence of minima. In this setting, we prove that the Palais-Smale condition for functionals bounded below implies the coerci…

We give sufficient conditions for a $C^1_c$-local diffeomorphism between Fréchet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fréchet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a functio…

The Palais-Smale condition is proven for various knot energies.

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 i…

The paper proves critical point results for Frechet manifolds.

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=…

Study Palais-Smale sequences for Ricci curvature on homogeneous spaces.

Variational method for eigenvalues on manifolds.

We describe the asymptotic behavior of Palais-Smale sequences associated to certain Yamabe-type equations on manifolds with boundary. We prove that each of those sequences converges to a solution of the limit equation plus a finite number of "bubbles" which are obtained by rescaling fundamental solutions of the corresp…

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 s…

We extend Lusternik-Schnirelmann theory to pairs $(f, φ)$, where $φ$ is a homotopy equivalence of a space $X$, $f$ is a function on $X$ which decreases along $φ$ and $(f, φ)$ satisfies a discrete analog of the Palais-Smale condition. The theory is carried out in an equivariant setting.

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.

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 tha…

In this paper we consider the functional whose critical points are solutions of the fractional CR Yamabe type equation on the sphere. We firstly study the behavior of the Palais-Smale sequences characterizing the bubbling phenomena and therefore we prove a multiplicity type result by showing the existence of infinitely…

In this paper we study the Palais-Smale sequences of the conformal Dirac-Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the existence of infinitely many solutions to the problem provided that the underlying mani…

We prove the Morse relations for the set of all geodesics connecting two non-conjugate points on a class of globally hyperbolic Lorentzian manifolds. We overcome the difficulties coming from the fact that the Morse index of every geodesic is infinite, and from the lack of the Palais-Smale condition, by using the Morse …

Proves existence of planar curves with specific curvature.

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 proo…

Using the reformulation in divergence form of the Euler-Lagrange equation for the Willmore functional as it was developed in "Analysis of the Willmore Functional" by T. Riviere (Invent. Math. 174), we study the limit of a local Palais-Smale sequence of weak Willmore immersions with locally square-integrable second fund…

Periodic geodesics on Hilbert half-Lie groups exist whenever the fundamental group is nontrivial.

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 Riem…

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 condi…

Study improves regularity estimates for harmonic maps into ellipsoids.

Ginzburg-Landau fields are the solutions of the Ginzburg-Landau equations which depend on two positive parameters, $α$ and $β$. We give conditions on $α$ and $β$ for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded …

We prove that the Yang-Mills $α$-functional satisfies the Palais-Smale condition. This guarantees the existence of critical points, which are called Yang-Mills $α$-connections. It was shown by Hong, Tian and Yin in [10] (to appear in Comm. Math. Helv.) that as $α\to 1$, a sequence of Yang-Mills $α$-connections converge…

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 asser…

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…

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…

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,…

We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface $Σ$ into a given closed manifold, we add to the area Lagrangian a term equal to the $L^q$ norm of the second fundamental form of the immersion times a "viscosity" parameter. …

For $n\ge 3$, let $Ω$ be a bounded domain in $R^n$ and $N$ be a compact Riemannian manifold in $R^L$ without boundary. Suppose that $u_n\in W^{1,n}(Ω,N)$ are the Palais-Smale sequences of the Dirichlet $n$-energy functional and $u_n$ converges weakly in $W^{1,n}$ to a map $u\in W^{1,n}(Ω,N)$. Then $u$ is a $n$-harmonic…

We prove the compactness of solutions to general fourth order elliptic equations which are L^1-perturbations of the Q-curvature equation on compact Riemannian 4-maniods. Consequently, we prove the global existence and convergence of the Q-curvature flow on a generic class of Riemannian 4-manifolds. As a by product, we …

New method generates critical points for complex functionals.

$α$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $α$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $α>1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in …

We introduce a general scheme that permits to generate successive min-max problems for producing critical points of higher and higher indices to Palais-Smale Functionals in Banach manifolds equipped with Finsler structures. We call the resulting tree of minmax problems a minmax hierarchy. Using the viscosity approach t…

Study of critical points for 4D conformally invariant curvature energies.

Optimizes metrics on surfaces for eigenvalues.

The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called $ES-r$-energy funct…

Given a manifold $M$ and a proper sub-bundle $Δ\subset TM$, we study homotopy properties of the horizontal base-point free loop space $Λ$, i.e. the space of absolutely continuous maps $γ:S^1\to M$ whose velocities are constrained to $Δ$ (for example: legendrian knots in a contact manifold). A key technical ingredient f…

Generates samples conditioned on labels using optimal transport.

The paper classifies Finsler surfaces satisfying the T-condition or σT-condition.

The paper develops a new approach to conditional risk measures using modular convex analysis.

Paper constructs solutions to Bogomolny equations with specific boundary and asymptotic conditions.

We extend probabilistic programming to handle conditioning on marginal distributions.

New tests for conditional copulas based on decision trees.

Paper finds necessary condition for logarithmic Minkowski problem in higher dimensions.

This paper introduces a neural operator for probabilistic conditioning.

CSI method learns conditional distributions by estimating flow equations.