arXiv research

Kuranishi's proof of complex deformation theory revisited

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

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

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.

Note on advancements in nonlinear elliptic equations' regularity theory.

The paper estimates curvature for a specific flow on manifolds.

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…

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

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

Study Poisson cohomology and linearize Lie algebra structures.

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

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…

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

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

Proves the Hodge conjecture for complex projective manifolds.

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

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.

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

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…

Constructs smooth integrable magnetic systems on a two-torus.

New Zoll families of minimal spheres found in spheres and projective 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…

Gradient estimates derived for solutions of a specific elliptic equation on Riemannian manifolds.

We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Na…

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

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

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

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 …

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 …

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

In this article we consider the motion of relativistic strings in the Minkowski space $\textbf{R}^{1+n}$. Those surfaces are known as a timelike minimal surface, and described by a system with $n$ nonlinear wave equations of Born-Infeld type. By constructing a suitable Nash-Moser iteration scheme, we prove that the $n$…

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…

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…

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…

We study a number of local and global classification problems in generalized complex geometry. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a …

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…

This paper studies the large time existence for the motion of closed hypersurfaces in a radially symmetric potential. In physical, this surface can be considered as an electrically charged membrane with a constant charge per area in a radially symmetric potential. The evolution of such surface has been investigated by …

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…

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…

Paper proves stability of multi-dimensional rarefaction waves in gas dynamics.

Alternative proof and extension of curvature estimates for minimal immersions.

This thesis studies normal forms for Poisson structures around symplectic leaves using several techniques: geometric, formal and analytic ones. One of the main results (Theorem 2) is a normal form theorem in Poisson geometry, which is the Poisson-geometric version of the Local Reeb Stability (from foliation theory) and…

If a differential equation in a Banach manifold is invariant or quasi-invariant under the action of one or more Lie groups, then its stationary points cannot be isolated, so that classical linearized stability theorem does not apply to it. The first main purpose of this paper is to establish a linearized stability theo…

Improved iterative methods for risk parity portfolio weights.

We describe a novel optimization method for finite sums (such as empirical risk minimization problems) building on the recently introduced SAGA method. Our method achieves an accelerated convergence rate on strongly convex smooth problems. Our method has only one parameter (a step size), and is radically simpler than o…