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Guillermo Henry

Publications and source records attributed to Guillermo Henry.

15 recordsLinked to original sources
arXiv2026

Interface foliation near a minimal isoparametric hypersurface for the Allen-Cahn equation

Let $(M,g)$ be a closed Riemannian manifold of positive Ricci curvature. Let $f$ be a proper isoparametric function on $M$, and $\Gamma$ the unique level set of $f$ which is a minimal hypersurface. For any positive integer $k$, and $\lambda >0$ large enough, we construct a solution of the Allen-Cahn equation $ \Delta u + \lambda (u-u^3 ) =0$ which is constant along the level sets of $f$ and has exactly $k$ nodal components. We prove that the nodal components approach the minimal isoparametric hypersurface $\Gamma$ as $\lambda \rightarrow \infty$ and the energy is uniformly bounded for all such solutions.

math.DG↗
arXiv2025

On solutions to Hardy-Sobolev equations on Riemannian manifolds

Let $(M,g)$ be a closed Riemannian manifold of dimension at least $3$. Let $S$ be the union of the focal submanifolds of an isoparametric function on $(M,g)$. In this article we address the existence of solutions of the Hardy-Sobolev type equation $\Delta_g u+K(x)u=\frac{u^{q-1}}{\left(d_{S}(x)\right)^s}$, where $d_{S}(x)$ is the distance from $x$ to $S$ and $q>2$. In particular, we will prove the existence of infinite sign-changing solutions to the equation.

math.AP↗
arXiv2023

Negative eingenvalues of the conformal Laplacian

Let $M$ be a closed differentiable manifold of dimension at least $3$. Let $Λ_0 (M)$ be the minimun number of non-positive eigenvalues that the conformal Laplacian of a metric on $M$ can have. We prove that for any $k$ greater than or equal to $Λ_0 (M)$, there exists a Riemannian metric on $M$ such that its conformal Laplacian has exactly $k$ negative eigenvalues. Also, we discuss upper bounds for $Λ_0 (M)$.

math.DG↗
arXiv2022

Isoparametric functions and solutions of Yamabe type equations on manifolds with boundary

Let $(M,g)$ be a compact Riemannian manifold with non-empty boundary. Provided $f$ an isoparametric function of $(M,g)$ we prove existence results for positive solutions of the Yamabe equation that are constant along the level sets of $f$. If $(M,g)$ has positive constant scalar curvature, minimal boundary and admits an isoparametric function we also prove multiplicity results for positive solutions of the Yamabe equation on $(M \times N,g+th) $ where $(N,h)$ is any closed Riemannian manifold with positive constant scalar curvature.

math.DG↗
arXiv2018

The Equivariant Second Yamabe Constant

For a closed Riemannian manifold of dimension $n\geq 3$ and a subgroup $G$ of the isometry group, we define and study the $G-$equivariant second Yamabe constant and we obtain some results on the existence of $G-$invariant nodal solutions of the Yamabe equation.

math.DG↗
arXiv2016

Second Yamabe Constant on Riemannian Products

Let $(M^m,g)$ be a closed Riemannian manifold $(m\geq 2)$ of positive scalar curvature and $(N^n,h)$ any closed manifold. We study the asymptotic behaviour of the second Yamabe constant and the second $N-$Yamabe constant of $(M\times N,g+th)$ as $t$ goes to $+\infty$. We obtain that $\lim_{t \to +\infty}Y^2(M\times N,[g+th])=2^{\frac{2}{m+n}}Y(M\times \re^n, [g+g_e]).$ If $n\geq 2$, we show the existence of nodal solutions of the Yamabe equation on $(M\times N,g+th)$ (provided $t$ large enough). When the scalar curvature of $(M,g)$ is constant, we prove that $\lim_{t \to +\infty}Y^2_N(M\times N,g+th)=2^{\frac{2}{m+n}}Y_{\re^n}(M\times \re^n, g+g_e)$. Also we study the second Yamabe invariant and the second $N-$Yamabe invariant.

math.DG↗
arXiv2016

On the instability of two entropic dynamical models

In this paper we study two entropic dynamical models from the viewpoint of information geometry. We study the geometry structures of the associated statistical manifolds. In order to analyse the character of the instability of the systems, we obtain their geodesics and compute their Jacobi vector fields. The results of this work improve and extend a recent advance in this topics studied in [13]

math-ph↗
arXiv2013

Isoparametric hypersurfaces and metrics of constant scalar curvature

We showed the existence of non-radial solutions of the equation $Δu -λu + λu^q =0$ on the round sphere $S^m$, for $q<2m/(m-2)$, and study the number of such solutions in terms of $λ$. We show that for any isoparametric hypersurface $M\subset S^m$ there are solutions such that $M$ is a regular level set (and the number of such solutions increases with $λ$). We also show similar results for isoparametric hypersurfaces in general Riemannian manifolds. These solutions give multiplicity results for metrics of constant scalar curvature on conformal classes of Riemannian products.

math.DG↗
arXiv2013

A note on Yamabe constants of products with hyperbolic spaces

We study the H^n-Yamabe constants of Riemannian products (H^n \times M^m, g_h^n +g), where (M,g) is a compact Riemannian manifold of constant scalar curvature and g_h^n is the hyperbolic metric on H^n. Numerical calculations can be carried out due to the uniqueness of (positive, finite energy) solutions of the equation Δu -λu + u^q =0 on hyperbolic space H^n under appropriate bounds on the parameters λ, q, as shown by G. Mancini and K. Sandeep. We do explicit numerical estimates in the cases (n,m)=(2,2),(2,3) and (3,2).

math.DG↗
arXiv2011

k-Nearest neighbor density estimation on Riemannian Manifolds

In this paper, we consider a k-nearest neighbor kernel type estimator when the random variables belong in a Riemannian manifolds. We study asymptotic properties such as the consistency and the asymptotic distribution. A simulation study is also consider to evaluate the performance of the proposal. Finally, to illustrate the potential applications of the proposed estimator, we analyzed two real example where two different manifolds are considered.

math.ST↗
arXiv2011

Robust Estimators in Partly Linear Regression Models on Riemannian Manifolds

Under a partially linear models we study a family of robust estimates for the regression parameter and the regression function when some of the predictor variables take values on a Riemannian manifold. We obtain the consistency and the asymptotic normality of the proposed estimators. Also, we consider a robust cross validation procedure to select the smoothing parameter. Simulations and application to real data show the performance of our proposal under small samples and contamination.

math.ST↗
arXiv2010

Partially linear models on Riemannian manifolds

In partially linear models the dependence of the response y on (x^T,t) is modeled through the relationship y=\x^T β+g(t)+ε where εis independent of (x^T,t). In this paper, estimators of βand g are constructed when the explanatory variables t take values on a Riemannian manifold. Our proposal combine the flexibility of these models with the complex structure of a set of explanatory variables. We prove that the resulting estimator of βis asymptotically normal under the suitable conditions. Through a simulation study, we explored the performance of the estimators. Finally, we applied the studied model to an example based on real dataset.

math.ST↗