arXiv research

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

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…

Variational method for eigenvalues on manifolds.

Paper studies fractional CR Yamabe equation on sphere, proving multiplicity of solutions.

Global diffeomorphism theorem for Fréchet spaces established.

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…

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…

Proves existence of planar curves with specific curvature.

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…

Study on Palais-Smale sequences for conformal Dirac-Einstein problem, proving existence of solutions.

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

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=…

The paper proves critical point results for Frechet 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…

The study finds conditions for irreducible Ginzburg-Landau fields on compact 2-manifolds.

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.

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.

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…

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…

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…

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…

Study improves regularity estimates for harmonic maps into ellipsoids.

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…

The paper explores higher order energy functionals and their critical points.

New method generates critical points for complex functionals.

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…

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 …

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 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…

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

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…

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 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 …

Generates critical points in manifolds for higher index min-max problems.

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. …

Optimizes metrics on surfaces for eigenvalues.

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

Paper proves existence of smooth nontrivial Dirac-harmonic maps.

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…

New neural network models for complex functional data analysis.

Knot signature function defined and conditions for its existence are given.

Distance function to a finite set is a topological Morse function.

Introduces new weighted floating functions and affine surface areas.

Develops methods for selecting and estimating smooth functional coefficients in high-dimensional multivariate functional data.

The paper introduces geodesic φ-convex functions and their properties.

Neural networks can approximate functionals on RKHS with error bounds.

FFBO optimizes functions as inputs and outputs, improving on existing BO methods.

Analyzes properties of transnormal Finsler functions on compact manifolds.