arXiv subjects

Jurgen Julio-Batalla

Publications and source records attributed to Jurgen Julio-Batalla.

9 recordsLinked to original sources

arXiv2026

Spinor inequality for magnetic fields on spin manifolds

This paper is concerned with the zero mode equation $D_gφ=iA\cdotφ$ on closed spin manifold $(M^n,g,σ)$ of positive scalar curvature. Here $A$ is a real one form on $M$. We proved that if $(φ, A)$ is a non trivial solution of the zero mode equation then $$\parallel dA\parallel_{n/2}>Y(M^n,[g])/(4v_n^{1/2}),$$ where $Y(M^n,[g])$ is the Yamabe constant of $(M^n,g)$ and $v_n=\left[\frac{n}{2}\right]$. In the case of the round sphere $(\mathbb{S}^n,g_{can},σ_{can})$ this result confirms that the inequality obtained in \cite{Frank} is not sharp.

math.DG↗

arXiv2026

Zero modes on product Riemannian manifolds

This paper is concerned with the zero mode equation $D_gφ=iA\cdotφ$ on product of closed spin manifolds $(M_1^{n_1}\times M_2^{n_2},g_1+g_2,σ)$ of dimensions $n_1\leq n_2$ respectively. Here $A$ is a real vector field on $M^n=M_1^{n_1}\times M_2^{n_2}$. Under non-increasing condition on $|φ|$ we prove that $$\parallel A\parallel_n^2\geq\frac{n_2}{4(n_2-1)}Y(M^n,[g]),$$ where $Y(M^n,[g])$ is the Yamabe constant of $(M^n,g)$. This estimate is sharp in even dimensions. We also obtain a similar estimate for non trivial solutions of the zero mode type equation $D_gφ=fφ$, where $f$ is a scalar function.

math.DG↗

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 $Δ^2 u -αΔu +βu = u^q$, where $α$ and $β$ are positive constants satisfying that $α^2 \geq 4 β$. 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↗

arXiv2024

Spinorial Yamabe-type equations and the Bär-Hijazi-Lott invariant

We consider on a closed Riemannian spin manifold $(M^n,g,σ)$ the spinorial Yamabe type equation $D_gφ=λ|φ|^{\frac{2}{n-1}}φ$, where $φ$ is a spinor field and $λ$ is a positive constant. For a normalized solution $φ$ of this equation we find a positive lower bound for $λ^2$. As an application we obtain an explicit lower bound of the Bär-Hijazi-Lott invariant for some spin manifolds with positive scalar curvature.

math.DG↗

arXiv2024

A note on sign-changing solutions to supercritical Yamabe-type equations

On a closed Riemannian manifold $(M^n ,g)$, we consider the Yamabe-type equation $-Δ_g u + λu = λ|u|^{q-1}u$, where $λ\in \mathbb{R}_{+}$ and $q>1$. We assume that $M$ admits a proper isoparametric function $f$ with focal submanifolds of positive dimension. If $k>0$ is the minimum of the dimensions of the focal submanifolds of $f$, we let $q^* =\frac{n-k+2}{n-k-2}$. We prove the existence of infinite $f$-invariant sign-changing solutions to the equation when $1<q<q^*$.

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↗

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

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

Isoparametric functions on $\mathbb{R}^n\times\mathbb{M}^m$

We classify the isoparametric functions on $\mathbb{R}^n\times\mathbb{M}^m$, $n, m\geq2$, with compact level sets, where $\mathbb{M}^m$ is a connected, closed Riemannian manifold of dimension $m$. Also, we classify the isoparametric hypersurfaces in $\mathbb{S}^2\times\mathbb{R}^2$ with constant principal curvatures.

math.DG↗