arXiv subjects

Jimmy Petean

Publications and source records attributed to Jimmy Petean.

At least 19 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↗

arXiv2024

Nodal solutions to Paneitz-type equations

On a closed Riemannian manifold $(M^n ,g)$ with a proper isoparametric function $f$ we consider the equation $\Delta^2 u -\alpha \Delta u +\beta u = u^q$, where $\alpha$ and $\beta$ are positive constants satisfying that $\alpha^2 \geq 4 \beta$. We let ${\bf m}$ be the minimum of the dimensions of the focal varieties of $f$ and $q_f = \frac{n-{\bf m}+4}{n-{\bf m}-4}$, $q_f = \infty$ if $n\leq {\bf m}+4$. We prove the existence of infinitely many nodal solutions of the equation assuming that $1<q<q_f$. The solutions are $f$-invariant. To obtain the result, first we prove a $C^0-$estimate for positive $f$-invariant solutions of the equation. Then we prove the existence of mountain pass solutions with arbitrarily large energy.

math.AP↗

arXiv2023

Global bifurcation for Paneitz type equations and constant Q-curvature metrics

We consider the Paneitz-type equation $Δ^2 u -αΔu +β(u-u^q ) =0$ on a closed Riemannian manifold $(M,g)$. We reduce the equation to a fourth-order ordinary differential equation assuming that $(M,g)$ admits a proper isoparametric function. Assuming that $α$ and $β$ are positive and $α^2 >4β$, we prove that the global nonconstant solutions of this ordinary differential equation only has nondegenerate critical points. Applying global bifurcation theory we prove multiplicity results for positive solutions of the equation. As an application and motivation we prove multiplicity results for conformal constant $Q$-curvature metrics. For example, consider closed positive Einstein manifolds $(M^n ,g )$ and $(X^m , h)$ of dimensions $n,m \geq 3$. Assuming that $M$ admits a proper isoparametric function (with a symmetry condition) we prove that as $δ>0$ gets closer to 0, the number of constant $Q$-curvature metrics conformal to $g_δ = g+δh$ goes to infinity.

math.DG↗

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↗

arXiv2023

Multiplicity results for constant Q-curvature conformal metrics

We prove that certain subcritical Paneitz-Branson type equations on a closed Riemannian manifold $(M,g)$ have at least $\mathrm{Cat}(M)$ positIve solutions, where $\mathrm{Cat}(M)$ is the Lusternik-Schnirelmann category of $M$. This implies that if $(X,h)$ is a closed positive Einstein manifold then for $\ep >0$ small enough there are at least $\mathrm{Cat}(M)$ metrics of constant $Q$-curvature in the conformal class of the Riemannian product $g+\ep h$.

math.DG↗

arXiv2022

Local bifurcation diagrams and degenerate solutions of Yamabe-type equations

We study positive solutions of the equation $-Δ_g u + λu = λu^q$, with $λ>0$, $q>1$ on the round sphere $\mathbb{S}^n$ . We reduce the equation to an ordinary differential equation by considering isoparametric functions and apply bifurcation theory. We study when the corresponding bifurcation points are transcritical. We apply this result to show the existence of degenerate solutions to the equation and to study multiplicity results for conformal constant scalar curvature metrics.

math.DG↗

arXiv2020

Nodal solutions of Yamabe-type equations on positive Ricci curvature manifolds

We consider a closed cohomogeneity one Riemannian manifold $(M^n,g) $ of dimension $n\geq 3$. If the Ricci curvature of $M$ is positive, we prove the existence of infinite nodal solutions for equations of the form $-Δ_g u + λu = λu^q$ with $λ>0$, $q>1$. In particular for a positive Einstein manifold which is of cohomogeneity one or fibers over a cohomogeniety one Einstein manifold we prove the existence of infinite nodal solutions for the Yamabe equation, with a prescribed number of connected components of its nodal domain.

math.DG↗

arXiv2019

Low energy nodal solutions to the Yamabe equation

Given an isoparametric function $f$ on the $n$-dimensional sphere, we consider the space of functions $w\circ f$ to reduce the Yamabe equation on the round sphere into a singular ODE on $w$ in the interval $[0,π]$, of the form $w" + (h(r)/\sin r)w'+λ(\vert w\vert^{4/n-2}w - w)=0$, where $h$ is a monotone function with exactly one zero on $[0,π]$ and $λ>0$ is a constant. The natural boundary conditions in order to obtain smooth solutions are $w'(0)=0$ and $w'(π)=0$. We show that for any positive integer $k$ there exists a solution with exactly $k$-zeroes yielding solutions to the Yamabe equation with exactly $k$ connected isoparametric hypersurfaces as nodal set. The idea of the proof is to consider the initial value problems on both singularities $0$ and $π$, and then to solve the corresponding double shooting problem, matching the values of $w$ and $w'$ at the unique zero of $h$. In particular we obtain solutions with exactly one zero, providing solutions of the Yamabe equation with low energy, which can be computed easily by numerical methods.

math.AP↗

arXiv2019

Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

Given an isoparametric function $f$ on the $n$-dimensional round sphere, we consider functions of the form $u=w\circ f$ to reduce the semilinear elliptic problem \[ -Δ_{g_0}u+λu=λ | u\ | ^{p-1}u\qquad\text{ on }\mathbb{S}^n \] with $λ>0$ and $1 \frac{n+2}{n-2}$, i.e., in the supercritical case. Moreover, using a reduction via harmonic morphisms, we prove existence and multiplicity of sign-changing solutions to the Yamabe problem on the complex and quaternionic space, having a finite disjoint union of isoparametric hipersurfaces as regular level sets.

math.AP↗

arXiv2019

Global bifurcation techniques for Yamabe type equations on Riemannian manifolds

We consider a closed Riemannian manifold $(M^n ,g)$ of dimension $n\geq 3$ and study positive solutions of the equation $-Δ_g u + λu = λu^q$, with $λ>0$, $q>1$. If $M$ supports a proper isoparametric function with focal varieties $M_1$, $M_2$ of dimension $d_1 \geq d_2 $ we show that for any $q<\frac{ n-d_2+2 }{n - d_2 -2}$ the number of positive solutions of the equation $-Δ_g u + λu = λu^q$ tends to $\infty$ as $λ\rightarrow +\infty$. We apply this result to prove multiplicity results for positive solutions of critical and supercritical equations. In particular we prove multiplicity results for the Yamabe equation on Riemannian manifolds.

math.DG↗

arXiv2018

A note on solutions of Yamabe-type equations on products of spheres

We consider Yamabe-type equations on the Riemannian product of constant curvature metrics on $\textbf{S}^n \times\textbf{ S}^n$, and study solutions which are invariant by the cohomogeneity one diagonal action of $O(n+1)$. We obtain multiplicity results for both positive and nodal solutions. In particular we prove the existence of nodal solutions of the Yamabe equation on these products which depend non-trivially on both factors

math.DG↗

arXiv2018

Bifurcation for the constant scalar curvature equation and harmonic Riemannian submersions

We study bifurcation for the constant scalar curvature equation along a one-parameter family of Riemannian metrics on the total space of a harmonic Riemannian submersion. We provide an existence theorem for bifurcation points and a criterion to see that the conformal factors corresponding to the bifurcated metrics must be indeed constant along the fibers. In the case of the canonical variation of a Riemannian submersion with totally geodesic fibers, we characterize discreteness of the set of all degeneracy points along the family and give a sufficient condition to guarantee that bifurcation necessarily occurs at every point where the linearized equation has a nontrivial solution. In the model case of quaternionic Hopf fibrations, we show that symmetry-breaking bifurcation does not occur except at the round metric.

math.DG↗

arXiv2016

Multiplicity results for the Yamabe equation by Lusternik-Schnirelmann theory

Let $(M,g)$ be any closed Riemannianan manifold and $(N,h)$ be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product $(M\times N , g + δh)$ has at least $Cat(M) +1 $ solutions for $δ$ small enough, where $Cat(M)$ denotes the Lusternik-Schnirelmann-category of $M$. Cat(M) of the solutions obtained have energy arbitrarily close to the minimum.

math.DG↗

arXiv2016

Metrics of constant scalar curvature on sphere bundles

Let $G/H$ be a Riemannian homogeneous space. For an orthogonal representation $ϕ$ of $H$ on the Euclidean space $\mathbb{R}^{k+1}$, there corresponds the vector bundle $E=G\times_ϕ\mathbb{R}^{k+1} \to G/H$ with fiberwise inner product. Provided that $ϕ$ is the direct sum of at most two representations which are either trivial or irreducible, we construct metrics of constant scalar curvature on the unit sphere bundle $UE$ of $E$. When $G/H$ is the round sphere, we study the number of constant scalar curvature metrics in the conformal classes of these metrics.

math.DG↗

arXiv2015

Stable solutions of the Yamabe equation on non-compact manifolds

We consider the Yamabe equation on a complete non-compact Riemannian manifold and study the condition of stability of solutions. If $(M^m,g)$ is a closed manifold of constant positive scalar curvature, which we normalize to be $m(m-1)$, we consider the Riemannian product with the $n$-dimensional Euclidean space: $(M^m \times \mathbf{R}^n, g+ g_E)$. And study the solution of the Yamabe equation which depends only on the Euclidean factor. We show that there exists a constant $λ(m,n)$ such that the solution is stable if and only if $λ_1 \geq λ(m,n)$, where $λ_1$ is the first positive eigenvalue of $-Δ_g$. We compute $λ(m,n)$ numerically for small values of $m,n$ showing in these cases that the Euclidean minimizer is stable in the case $M=S^m $ with the metric of constant curvature. This implies that the same is true for any closed manifold with a Yamabe metric.

math.DG↗

arXiv2014

Minimal hypersurfaces in R^n \times S^m

We classify minimal hypersurfaces in $R^n \times S^m$, $n,m \geq 2$, which are invariant by the canonical action of $O(n) \times O(m)$. We also construct compact and noncompact examples of invariant hypersurfaces of constant mean curvature. We show that the minimal hypersurfaces and the noncompact constant mean curvature hypersurfaces are all unstable.

math.DG↗

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↗