arXiv research

The paper constructs metrics on spheres with families of minimal hypersurfaces.

Proves the Hodge conjecture for complex projective manifolds.

Kuranishi's proof of complex deformation theory revisited

Constructs smooth integrable magnetic systems on a two-torus.

This article provides a version of scale calculus geared towards a notion of (nonlinear) Fredholm maps between certain types of Frechet spaces, retaining as many as possible of the properties Fredholm maps between Banach spaces enjoy, and the existence of a constant rank theorem for such maps. It does so by extending t…

Harmonic functions stable under small changes.

Study on a metric on Hermitian metrics space, proving diffeomorphisms and completeness.

Second part of proving linearization theorem for sl2(C).

Study metric perturbations to make degenerate harmonic forms non-degenerate.

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse fun…

The paper establishes gradient estimates and Liouville theorems for Φ-Laplacian equations on Riemannian manifolds.

A geometric flow on $(2,2)$-forms is introduced which preserves the balanced condition of metrics, and whose stationary points satisfy the anomaly equation in Strominger systems. The existence of solutions for a short time is established, using Hamilton's version of the Nash-Moser implicit function theorem.

The paper uses Nash-Moser iteration to prove gradient estimates for nonlinear equations on Riemannian manifolds.

The paper studies gradient estimates for solutions of a nonlinear elliptic equation on Riemannian manifolds.

Sharp gradient estimates for a weighted p-Laplacian equation on metric measure spaces.

Let (N,g) be a Riemannian manifold. For a compact, connected and oriented submanifold M of N. we define the space of volume preserving embeddings Emb_μ(M,N) as the set of smooth embeddings f:M \rightarrow N such that f*μ^{f}=μ, where μ^{f} (resp. μ) is the Riemannian volume form on f(M) (resp. M) induced by the ambient…

The article studies deformations of $\mathbb Z_2$-harmonic spinors on 3-manifolds.

Paper analyzes solutions to quasilinear elliptic equations on manifolds using Nash-Moser iteration.

This paper is about the rigidity of compact group actions in the Poisson context. The main resut is that Hamiltonian actions of compact semisimple type are rigid. We prove it via a Nash-Moser normal form theorem for closed subgroups of SCI-type. This Nash-Moser normal form has other applications to stability results th…

We apply the Nash-Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space $H_{k-nil}^2(\math…

We derive a Harnack inequality for positive solutions of the $f$-heat equation and Gaussian upper and lower bounds for the $f$-heat kernel on complete smooth metric measure spaces $(M, g, e^{-f}dv)$ with Bakry-Émery Ricci curvature bounded below. The lower bound is sharp. The main argument is the De Giorgi-Nash-Moser t…

Global inverse function theorem proved easily using Riemannian geometry.

Study Poisson cohomology and linearize Lie algebra structures.

We give a local classification of generalized complex structures. About a point, a generalized complex structure is equivalent to a product of a symplectic manifold with a holomorphic Poisson manifold. We use a Nash-Moser type argument in the style of Conn's linearization theorem.

Constructs harmonic spinors and 1-forms on 3-manifold connected sums and torus sums.

Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.

The paper proves a generalized inverse function theorem for curved $L_\infty$ spaces.

We prove the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids near a singular point. In the appendix of this paper, we show an abstract Nash-Moser normal form theorem, which generalizes our Levi decomposition result and which may be helpful in the study of other smooth nor…

This thesis is divided into two parts. In the first part we study completely integrable systems, and their underlying structures, in detail. We study their deformation theory and the different equivalence relations surrounding it. We motivate the definition of weak equivalence (found in the literature) by studying diff…

We prove the rigidity of presymplectic actions of a compact semisimple Lie algebra on a presymplectic manifold of constant rank in the local and global case. The proof uses an abstract normal form theorem we had stated in a previous work, based on an iterative process of Nash-Moser type. In order to use correctly this …

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash-Moser fast convergence method. In the case of one-point submanifolds (fixed points), this immediately implies a stronger version of Conn's linearization theorem, also proving that Conn's theorem is, indeed, just a manife…

The paper provides new gradient estimates for solutions to a nonlinear elliptic equation on smooth metric measure spaces.

The Global Newlander-Nirenberg theorem is proven for domains with finite smooth boundary in complex manifolds.

New Zoll families of minimal spheres found in spheres and projective spaces.

Abstracts a theorem for non-smooth maps in infinite dimensions.

Inverse function theorem and homotopy description for L-infinity bundles.

The Stone-Weierstrass theorem aids in solving inverse problems on specific manifolds.

Note on advancements in nonlinear elliptic equations' regularity theory.

The paper details local forms of morphisms in colored supermanifolds.

A general slice theorem for the action of a Fréchet Lie group on a Fréchet manifolds is established. The Nash-Moser theorem provides the fundamental tool to generalize the result of Palais to this infinite-dimensional setting. The presented slice theorem is illustrated by its application to gauge theories: the action o…

Represents neural networks as solutions to inverse problems in Banach spaces.

In this paper, we give a proof of the quantitative Morse theorem stated by {Y. Yomdin} in \cite{Y1}. The proof is based on the quantitative Sard theorem, the quantitative inverse function theorem and the quantitative Morse lemma.

New findings on mesh group-planes validate Signature-inverse Theorem under specific conditions.

Building upon ideas of Hironaka, Bierstone-Milman, Malgrange and others we generalize the inverse and implicit function theorem (in differential, analytic and algebraic setting) to sets of functions of larger multiplicities (or ideals). This allows one to describe singularities given by a finite set of generators or by…

We develop a comprehensive study on sharp potential type Riemannian Sobolev inequalities of order 2 by means of a local geometric Sobolev inequality of same kind and suitable De Giorgi-Nash-Moser estimates. In particular we discuss questions like continuous dependence of optimal constants and existence and compactness …

The paper estimates curvature for a specific flow on manifolds.

Researchers solve a complex equation to embed graphs with negative curvature.

We study neighbourhoods of submanifolds in generalized complex geometry. Our first main result provides sufficient criteria for such a submanifold to admit a neighbourhood on which the generalized complex structure is B-field equivalent to a holomorphic Poisson structure. This is intimately tied with our second main re…