arXiv research

We extend a work of Bartsch, Clapp and Puppe on the Mountain pass theorems. We consider functionals invariant with respect to infinite discrete groups satisfying a maximality condition on the finite subgroups.

Proves uniqueness of small entropy self-expanders.

New theorem finds new minimal hypersurfaces in hyperbolic space.

Study on nonlinear elliptic equations with variable exponents, proving existence and multiplicity of solutions.

We provide sufficient conditions for the existence of a global diffeomorphism between tame Fréchet spaces. We prove a version of the Mountain Pass Theorem which is a key ingredient in the proof of the main theorem.

Study finds critical points in perimeter functional for fixed volume sets.

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…

The paper proves critical point results for Frechet manifolds.

New self-expander found between two given asymptotic ones.

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…

In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold $γ$ contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound $γ$. In order to do so, we deve…

Study on ground states of semilinear elliptic equations with various potential wells.

Two methods find at least two solutions to Kazdan-Warner's problem on surfaces.

The paper proves the existence of infinitely many nodal solutions to a Paneitz-type equation.

In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 2$. We develop a direct variational methods similar to the proof of the famous Plateau problem by J. Douglas and T. Rado. As a result, we show that the min-max value for the area functional can be achieved by a …

The paper proves the existence of hypersurfaces with prescribed mean curvature.

We construct and analyze minimal disc stackings with bounds on their Morse index.

New theorem connects distant points and identical points on manifolds.

We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, i…

We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem without Stabilization. We establish the existence of a nontrivial knot-type specif…

The paper proves the existence of finite-energy solutions to a specific elliptic equation on Riemannian manifolds.

Study the landscape of Lipschitz functions between manifolds using persistent homology.

Is it possible to obtain unbounded minimal surfaces in certain asymptotically flat 3-manifolds as a limit of solutions to a natural mountain pass problem with diverging boundaries? In this work, we give evidence that this might be true by analyzing related aspects in the case of the exact Riemannian Schwarzschild manif…

Solves time-minimizing navigation on a mountain slope using Riemann-Finsler geometry.

For a stable marginally outer trapped surface (MOTS) in an axially symmetric spacetime with cosmological constant $Λ> 0$ and with matter satisfying the dominant energy condition, we prove that the area $A$ and the angular momentum $J$ satisfy the inequality $8π|J| \le A\sqrt{(1-ΛA/4π)(1-ΛA/12π)}$ which is saturated pre…

The geometry on a slope of a mountain is the geometry of a Finsler metric, called here the {\it slope metric}. We study the existence of globally defined slope metrics on surfaces of revolution as well as the geodesic's behavior. A comparison between Finslerian and Riemannian areas of a bounded region is also studied.

Study examines diversification of mid-mountain ski tourism.

Post-pandemic, work patterns shifted with fewer days in offices and a new midweek mountain.

Study improves precipitation predictions for High Mountain Asia using machine learning.

Proves existence of a single-valued minimal hypersurface in compact manifolds.

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. …

Solves time-optimal navigation on slippery slopes with cross gravitational wind.

In this paper, we applied the multifractal detrended fluctuation analysis to the daily means of wind speed measured by 119 weather stations distributed over the territory of Switzerland. The analysis was focused on the inner time fluctuations of wind speed, which could be more linked with the local conditions of the hi…

The goal of this paper is to describe all local diffeomorphisms mapping a family of circles, in an open subset of $\r^3$, into straight lines. This paper contains two main results. The first is a complete description of the rectifiable collection of circles in $\r^3$ passing through one point. It turns out that to be r…

Bayesian online learning algorithm for one-pass data, achieving frequentist validity and uncertainty quantification.

We prove that the canonical 4-dimensional surgery problems can be solved after passing to a double cover. This contrasts the long-standing conjecture about the validity of the topological surgery theorem for arbitrary fundamental groups (without passing to a cover). As a corollary, the surgery conjecture is reformulate…

It is shown that two braids represent transversally isotopic links if and only if one can pass from one braid to another by conjugations in braid groups, positive Markov moves, and their inverses.

We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the the fixed point case (known as Zung's theorem) we give a shorter and more geometric proof, based on a Moser deformation argument. The passing to general orbits (Weinstein) is given a more conceptual inte…

For non-homotopic maps $u,v\in C^{\infty}(M,N)$ between closed Riemannian manifolds, we consider the smallest energy level $γ_p(u,v)$ for which there exist paths $u_t\in W^{1,p}(M,N)$ connecting $u_0=u$ to $u_1=v$ with $\|du_t\|_{L^p}^p\leq γ_p(u,v)$. When $u$ and $v$ are $(k-2)$-homotopic, work of Hang and Lin shows t…

We develop a method of extending actions of compact transformation groups which is then applied to the problem of preservation of equivariant extensor property by passing to a subspace of given orbit types.

Consider a smooth map from a neighborhood of the origin in a real vector space to a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this …

The abstract proves a global splitting theorem for Poisson manifolds.

New examples show causality conditions don't always pass to coverings.

NAST generalizes scattering transform for non-stationary time series analysis.

Introduces LoCA regret to evaluate model-based RL methods.

The Kinoshita graph is a particular embedding in the 3-sphere of a graph with three edges, two vertices and no loops. It has the remarkable property that although the removal of any edge results in an unknotted loop, the Kinoshita graph is itself knotted. We use two classical theorems from knot theory to give two parti…

SHINE uses forward pass quasi-Newton matrices to approximate Jacobian inverses for faster bi-level optimization.

We simplify proof of the theorem that close to any pseudoholomorphic disk there passes a pseudoholomorphic disk of arbitrary close size with any pre-described sufficiently close direction. We apply these results to the Kobayashi and Hanh pseudodistances. It is shown they coincide in dimensions higher than four. The res…