arXiv subjects

Juan Miguel Ruiz

Publications and source records attributed to Juan Miguel Ruiz.

12 recordsLinked to original sources

arXiv2026

On K-peak solutions for the Yamabe equation on product manifolds

Let $(M^n, g)$ and $(X^m, h)$ be closed manifolds $m, n>2$, such that $(X, h)$ has constant positive scalar curvature. We consider the one parameter family of products $(M\times X, g+ε^2 h)$, $ε>0$. We prove that if either the scalar curvature of $g$, $s_g$, is constant or a certain dimensional constant $β=0$, there is some function $Φ:M\rightarrow \mathbb{R}$ that depends on $s_g$, the norm of the Ricci curvature of $g$ and the norm of the curvature tensor of $g$; such that if $ξ_0$ is a stable, isolated, critical point of $Φ$, then for each $K\in\mathbb{N}$, there is some $ε_0>0$ such that for every $ε\in (0,ε_0)$ the subcritical Yamabe equation $-ε^2Δ_g u+(1+{\bf{c}}ε^2 s_g)u=u^q$ has a positive $K-$peak solution, which concentrates around $ξ_0$. Here, ${\bf{c}}=\frac{N-2}{4(N-1)}$, $q=\frac{N+2}{N-2}$ and $N=n+m$. This provides solutions for the Yamabe equation on Riemannian products $(M\times X, g+ε^2 h)$ and covers some remaining cases of previous results which handle the case where $s_g$ has non-degenerate critical points and $β\neq0$.

math.DG↗

arXiv2026

Isoperimetric estimates in the product of small and large volume manifolds

Let $(M^m,g)$, $(N^n,h)$ be closed Riemannian manifolds, $m,n\geq 2$, with concave isoperimetric profiles and volumes $V_M$, $V_N$ respectively. We consider a one parameter family of product manifolds of the same volume, $(X,G_λ)=(M^m\times N^n,λ^{2n}g+ λ^{-2m}h)$, $λ>0$, and estimate a lower bound for their isoperimetric profile for big $λ$. In particular, we show that for $α\in (\frac{3}{4},1)$ and $v_0 \in (0, V_MV_N)$, there is some $λ_{0}>0$, such that for $λ>λ_0$, we can bound the isoperimetric profile of $(X,G_λ)$: $$ α^{4} f_{M,λ}(v_0) \leq I_{(X,G_λ)}(v_0)\leq f_{M,λ}(v_0)$$ where $f_{M,λ}(v)= λ^{-n} V_N I_{(M,g)}(\frac{v}{V_N})$ and $ I_{(M,g)}$ is the isoperimetric profile of $(M,g)$. Moreover if $(M,g)=(S^m,g_0)$, the $m-$sphere with the round metric, in this setting, we show that some regions of the type ${ D^λ(r)\times N_λ }$, are actual isoperimetric regions in $ (S^m\times N^n,λ^{2n}g_0+ λ^{-2m}h)$ when $λ$ is big enough; being $D^λ(r)$ a disk on $(S^m,λ^{2n}g_0)$ and $N_λ=(N, λ^{-2m}h)$.

math.DG↗

arXiv2023

Lower bounds for isoperimetric profiles and Yamabe constants

We estimate explicit lower bounds for the isoperimetric profiles of the Riemannian product of a compact manifold and the Euclidean space with the flat metric, $(M^m\times \mathbb{R}^n,g+g_E)$, $m,n>1$. In particular, we introduce a lower bound for the isoperimetric profile of $M^m\times \mathbb{R}^n$ for regions of large volume and we improve on previous estimates of lower bounds for the isoperimetric profiles of $S^2 \times \mathbb{R}^2$, $S^3 \times \mathbb{R}^2$, $S^2 \times \mathbb{R}^3$. We also discuss some applications of these results in order to improve known lower bounds for the Yamabe invariant of certain manifolds.

math.DG↗

arXiv2020

Isoperimetric estimates in low dimensional Riemannian products

Let $(T^k,h_k)=(S_{r_1}^1\times S_{r_2}^1 \times ... \times S_{r_k}^1, dt_1^2+dt_2^2+...+dt_k^2)$ be flat tori, $r_k\geq ...\geq r_2\geq r_1>0$ and $(\mathbb R^n,g_E)$ the Euclidean space with the flat metric. We compute the isoperimetric profile of $(T^2\times \mathbb R^n, h_2+g_E)$, $2\leq n\leq 5$, for small and big values of the volume. These computations give explicit lower bounds for the isoperimetric profile of $T^2\times\mathbb R^n$. We also note that similar estimates for $(T^k\times \mathbb R^n, h_k+g_E)$, $2\leq k\leq5$, $2\leq n\leq 7-k$, may be computed, provided estimates for $(T^{k-1}\times \mathbb R^{n+1}, h_{k-1}+g_E)$ exist. We compute this explicitly for $k=3$. We use symmetrization techniques for product manifolds, based on work of A. Ros and F. Morgan.

math.DG↗

arXiv2018

Multipeak solutions for the Yamabe equation

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$ and $x_0 \in M$ be an isolated local minimum of the scalar curvature $s_g$ of $g$. For any positive integer $k$ we prove that for $ε>0$ small enough the subcritical Yamabe equation $-ε^2 Δu +(1+ c_{N} \ ε^2 s_g ) u = u^q$ has a positive $k$-peaks solution which concentrate around $x_0$, assuming that a constant $β$ is non-zero. In the equation $c_N = \frac{N-2}{4(N-1)}$ for an integer $N>n$ and $q= \frac{N+2}{N-2}$. The constant $β$ depends on $n$ and $N$, and can be easily computed numerically, being negative in all cases considered. This provides solutions to the Yamabe equation on Riemannian products $(M\times X , g+ ε^2 h )$, where $(X,h)$ is a Riemannian manifold with constant positive scalar curvature. We also prove that solutions with small energy only have one local maximum.

math.AP↗

arXiv2016

Multiplicity of constant scalar curvature metrics on warped products

Let $(M^m,g_M)$ be a closed, connected manifold with positive scalar curvature and $(T^k,g)$ some flat $k$-Torus of unit volume. By a result of F. Dobarro and E. Lami Dozo, there exists a unique $f: M \rightarrow \mathbf{R}_{>0}$ such that the warped product $M\times_f T^k$ has constant scalar curvature and unit volume. We study the Yamabe equation on these spaces. We use techniques from bifurcation theory, along with spectral theory for warped products, to prove multiplicity results for the Yamabe problem.

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↗

arXiv2012

On the Yamabe constants of $S^2 \times \re^3$ and $S^3 \times \re^2$

We compare the isoperimetric profiles of $S^2 \times \re^3$ and of $S^3 \times \re^2$ with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of $S^2 \times \re^3$ and $S^3 \times \re^2$. Explicitly we show that $Y(S^3 \times \re^2, [g_0^3 +dx^2]) > (3 /4) Y(S^5)$ and $Y(S^2 \times \re^3, [g_0^2 +dx^2]) > 0.63 Y(S^5)$. We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in previous work and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions.

math.DG↗

arXiv2010

Isoperimetric profile comparisons and Yamabe constants

We estimate from below the isoperimetric profile of $S^2 \times \re^2$ and use this information to obtain lower bounds for the Yamabe constant of $S^2 \times \re^2$. This provides a lower bound for the Yamabe invariants of products $S^2 \times M^2$ for any closed Riemann surface $M$. Explicitly we show that $Y(S^2 \times M^2) > (2/3) Y(S^4)$.

math.DG↗

arXiv2009

Results on the existence of the Yamabe minimizer of M^m \times R^n

We let (M^m, g) be a closed smooth Riemannian manifold (m >1) with positive scalar curvature S_g, and prove that the Yamabe constant of (M \times R^n,g+g_E) is achieved by a metric in the conformal class of (g+g_E), where g_E is the Euclidean metric. We also show that the Yamabe quotient of (M \times R^n,g+g_E) is improved by Steiner symmetrization with respect to M. It follows from this last assertion that the dependence on R^n of the Yamabe minimizer of (M \times R^n,g+g_E) is radial.

math.DG↗

arXiv2008

On metrics of positive Ricci curvature conformal to MxR^m

Let (M, g) be a closed Riemannian manifold and gE the Euclidean metric. We show that for m > 1, (M x R^m, (g + gE)) is not conformal to a positive Einstein manifold. Moreover, (M x R^m, (g + gE)) is not conformal to a Riemannian manifold of positive Ricci curvature, through a smooth, radial, positive, integrable function of R^m, for m > 1. These results are motivated by some recent questions on Yamabe constants.

math.DG↗