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Isabel M. C. Salavessa

Publications and source records attributed to Isabel M. C. Salavessa.

18 recordsLinked to original sources
arXiv2017

Heinz mean curvature estimates in warped product spaces $M\times_{e^ψ}N$

If a graph submanifold $(x,f(x))$ of a Riemannian warped product space $(M^m\times_{e^ψ}N^n,\tilde{g}=g+e^{2ψ}h)$ is immersed with parallel mean curvature $H$, then we obtain a Heinz type estimation of the mean curvature. Namely, on each compact domain $D$ of $M$, $m\|H\|\leq \frac{A_ψ(\partial D)}{V_ψ(D)}$ holds, where $A_ψ(\partial D)$ and $V_ψ(D)$ are the $ψ$-weighted area and volume, respectively. In particular, $H=0$ if $(M,g)$ has zero weighted Cheeger constant, a concept recently introduced by D.\ Impera et al.\ (\cite{[Im]}). This generalizes the known cases $n=1$ or $ψ=0$. We also conclude minimality using a closed calibration, assuming $(M,g_*)$ is complete where $g_*=g+e^{2ψ}f^*h$, and for some constants $α\geq δ\geq 0$, $C_1>0$ and $β\in [0,1)$, $\|\nabla^*ψ\|^2_{g_*}\leq δ$, $\mathrm{Ricci}_{ψ,g_*}\geq α$, and $det_g(g_*)\leq C_1 r^{2β}$ holds when $r\to +\infty$, where $r(x)$ is the distance function on $(M,g_*)$ from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.

math.DG↗
arXiv2012

Stable 3-spheres in $\mathbb{C}^3$

By only using spectral theory of the Laplace operator on spheres, we prove that the unit 3-dimensional sphere of a 2-dimensional complex subspace of $\mathbb{C}^3$ is a $Ω$-stable submanifold with parallel mean curvature, when $Ω$ is the Kähler calibration of rank 4 of $\mathbb{C}^3$.

math.DG↗
arXiv2011

Cauchy-Riemann inequalities on 2-spheres of $\mathbb{R}^7$

We prove that an integral Cauchy-Riemann inequality holds for any pair of smooth functions $(f,h)$ on the 2-sphere $\mathbb{S}^2$, and equality holds iff $f$ and $h$ are related $λ_1$-eigenfunctions. We extend such inequality to 4-tuples of functions, only valid on the $L^2$-orthogonal complement of a suitable nonzero finite dimensional space of functions. As a consequence we prove that 2-spheres are not $Ω$-stable surfaces with parallel mean curvature in $\mathbb{R}^7$ for the associative calibration $Ω$.

math.DG↗
arXiv2010

Stability of submanifolds with parallel mean curvature in calibrated manifolds

On a Riemannian manifold $\bar{M}^{m+n}$ with an $(m+1)$-calibration $Ω$, we prove that an $m$-submanifold $M$ with constant mean curvature $H$ and calibrated extended tangent space $\mathbb{R}H\oplus TM$ is a critical point of the area functional for variations that preserve the enclosed $Ω$-volume. This recovers the case described by Barbosa, do Carmo and Eschenburg, when $n=1$ and $Ω$ is the volume element of $\bar{M}$. To the second variation we associate an $Ω$-Jacobi operator and define $Ω$-stablility. Under natural conditions, we prove that the Euclidean $m$-spheres are the unique $Ω$-stable submanifolds of $\mathbb{R}^{m+n}$. We study the $Ω$-stability of geodesic $m$-spheres of a fibred space form $M^{m+n}$ with totally geodesic $(m+1)$-dimensional fibres.

math.DG↗
arXiv2010

Mean Curvature Flow of Spacelike Graphs

We prove the mean curvature flow of a spacelike graph in $(Σ_1\times Σ_2, g_1-g_2)$ of a map $f:Σ_1\to Σ_2$ from a closed Riemannian manifold $(Σ_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(Σ_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2\leq K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption $K_2\leq K_1$, that if $K_1>0$, or if $Ricci_1>0$ and $K_2\leq -c$, $c>0$ constant, any map $f:Σ_1\to Σ_2$ is trivially homotopic provided $f^*g_2<ρg_1$ where $ρ=\min_{Σ_1}K_1/\sup_{Σ_2}K_2^+\geq 0$, in case $K_1>0$, and $ρ=+\infty$ in case $K_2\leq 0$. This largely extends some known results for $K_i$ constant and $Σ_2$ compact, obtained using the Riemannian structure of $Σ_1\times Σ_2$, and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one.

math.DG↗
arXiv2009

Graphic Bernstein Results in Curved Pseudo-Riemannian Manifolds

We generalize a Bernstein-type result due to Albujer and Alías, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $Σ_1\times \mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete spacelike graphic submanifold with parallel mean curvature, defined by a map $f: Σ_1\to Σ_2$ between two Riemannian manifolds $(Σ_1^m, g_1)$ and $(Σ^n_2, g_2)$ of sectional curvatures $K_1$ and $K_2$, respectively. We take on $Σ_1\times Σ_2$ the pseudo-Riemannian product metric $g_1-g_2$. Under the curvature conditions, $\mathrm{Ricci}_1 \geq 0$ and $K_1\geq K_2$, we prove that, if the second fundamental form of $M$ satisfies an integrability condition, then $M$ is totally geodesic, and it is a slice if $\mathrm{Ricci}_1(p)>0$ at some point. For bounded $K_1$, $K_2$ and hyperbolic angle $θ$, we conclude $M$ must be maximal. If $M$ is a maximal surface and $K_1\geq K_2^+$, we show $M$ is totally geodesic with no need for further assumptions. Furthermore, $M$ is a slice if at some point $p\in Σ_1$, $K_1(p)> 0$, and if $Σ_1$ is flat and $K_2<0$ at some point $f(p)$, then the image of $f$ lies on a geodesic of $Σ_2$.

math.DG↗
arXiv2009

Bernstein-Heinz-Chern results in calibrated manifolds

Given $(\bar{M},Ω)$ a calibrated Riemannian manifold with a parallel calibration of rank $m$, and $M^m$ an immersed orientable submanifold with parallel mean curvature $H$ we prove that if $\cos θ$ is bounded away from zero, where $θ$ is the $Ω$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. In the particular case $M$ is complete with $Ricc^M\geq 0$ we may replace the boundedness condition on $\cos θ$ by $\cos θ\geq Cr^{-β}$, when $r\to +\infty$, where $ 0\leqβ<1 $ and $C > 0$ are constants and $r$ is the distance function to a point in $M$. Our proof is surprisingly simple and extends to a very large class of submanifolds in calibrated manifolds, in a unified way, the problem started by Heinz and Chern of estimating the mean curvature of graphic hypersurfaces in Euclidean spaces. It is based on a estimation of $\|H\|$ in terms of $\cosθ$ and an isoperimetric inequality. We also give some conditions to conclude $M$ is totally geodesic. We study some particular cases.

math.DG↗
arXiv2009

On the spectrum of $\bar{X}$-bounded minimal submanifolds

We prove, under a certain boundedness condition at infinity on the $(\bar{X}^{\top}, \bar{X}^{\bot})$-component of the second fundamental form, the vanishing of the essential spectrum of a complete minimal $\bar{X}$-bounded and $\bar{X}$-properly immersed submanifold on a Riemannian manifold endowed with a strongly convex vector field $\bar{X}$. The same conclusion also holds for any complete minimal $h$-bounded and $h$-properly immersed submanifold that lies in a open set of a Riemannian manifold $\oM$ supporting a nonnegative strictly convex function $h$. This extends a recent result of Bessa, Jorge and Montenegro on the spectrum of Martin-Morales minimal surfaces. Our proof uses as main tool an extension of Barta's theorem given in \cite{BM}

math.DG↗
arXiv2007

Spacelike graphs with parallel mean curvature

We consider spacelike graphs $Γ_f$ of simple products $(M\times N, g\times -h)$ where $(M,g)$ and $(N,h)$ are Riemannian manifolds and $f:M\to N$ is a smooth map. Under the condition of the Cheeger constant of $M$ to be zero and some condition on the second fundamental form at infinity, we conclude that if $Γ_f \subset M\times N$ has parallel mean curvature $H$ then $H=0$. This holds trivially if $M$ is closed. If $M$ is the $m$-hyperbolic space then for any constant $c$, we describe a explicit foliation of $H^m\times R$ by hypersurfaces with constant mean curvature $c$.

math.DG↗
arXiv2006

Power Mean Curvature Flow in Lorentzian Manifolds

We study the motion of an $n$-dimensional closed spacelike hypersurface in a Lorentzian manifold in the direction of its past directed normal vector, where the speed equals a positive power $p$ of the mean curvature. We prove that for any $p\in (0,1]$, the flow exists for all time when the Ricci tensor of the ambient space is bounded from below on the set of timelike unit vectors. Moreover, if we assume that all envolving hypersurfaces stay in a precompact region, then the flow converges to a stationary maximum spacelike hypersurface.

math.DG↗
arXiv2006

Cayley submanifolds of Calabi-Yau 4-folds

Our main results are: (1) The complex a Lagrangian points of a non-complex Lagrangian $2n$-dimensional submanifold $F:M\ra N$, immersed with parallel mean curvature and with equal Kaehler angles into a Kaehler-Einstein manifold $(N,J,g)$ of complex dimension $2n$, are zeros of finite order of $\sin^2θ$ and $\cos^2θ$ respectively, where $θ$ is the common $J$-Kaelher angle. (2) If $M$ is a Cayley submanifold of a Calabi-Yau (CY) manifold $N$ of complex dimension 4, then $\bigwedge^2_+NM$ is naturally isomorphic to $\bigwedge^2_+TM$. (3) If $N$ is Ricci-flat (not necessarily CY) and $M$ is a Cayley submanifold, then $ p_1(\bigwedge^2_+NM)= p_1(\bigwedge^2_+TM)$ still holds, but $p_1(\bigwedge^2_-NM)- p_1(\bigwedge^2_-TM)$ may describe a residue on the $J$-complex points, in the sense of Harvey and Lawson. We describe this residue by a PDE on a natural morphism $Φ:TM \to NM$, $Φ(X)=(JX)^{\bot}$, with singularities at the complex points. We give an explicit formula of this residue in a particular case. When $(N,I,J,K,g)$ is an hyper-Kaehler manifold and $M$ is an $I$-complex closed 4-submanifold, the first Weyl curvature invariant of $M$ may be described as a residue on the $J$-Kaehler angle at the $J$_Lagrangian points by a Lelong-Poincaré type formula. We study the almost complex structure $\Jw$ on $M$ induced by $F$.

math.DG↗
arXiv2006

Transgression forms in dimension 4

We compute explicit transgression forms for the Euler and Pontrjagin classes of a Riemannian manifold $M$ of dimension 4 under a conformal change of the metric, or a change to a Riemannian connection with torsion. These formulae describe the singular set of some connections with singularities on compact manifolds as a residue formula in terms of a polynomial of invariants. We give some applications for minimal submanifolds of Kaehler manifolds. We also express the difference of the first Chern class of two almost complex structures, and in particular an obstruction to the existence of a homotopy between them, by a residue formula along the set of anti-complex points. Finally, we take the first steps in the study of obstructions for two almost quaternionic-Hermitian structures on a manifold of dimension 8 to have homotopic fundamental forms or isomorphic twistor spaces.

math.DG↗
arXiv2005

On the twistor space of pseudo-spheres

We give a new proof that the sphere S^6 does not admit an integrable orthogonal complex structure, as in \cite{LeBrun}, following the methods from twistor theory. We present the twistor space of a pseudo-sphere S^{2n}_{2q}=SO_{2p+1,2q}/SO_{2p,2q} as a pseudo-Kähler symmetric space. We then consider orthogonal complex structures on the pseudo-sphere, only to prove such a structure cannot exist.

math.DG↗
arXiv2002

On the Kaehler angles of submanifolds

We prove that under certain conditions on the mean curvature and on the Kaehler angles, a compact submanifold M of real dimension 2n, immersed into a Kaehler-Einstein manifold N of complex dimension 2n, must be either a complex or a Lagrangian submanifold of N, or have constant Kaehler angle, depending on n=1, n=2, or n \geq 3, and the sign of the scalar curvature of N. These results generalize to non-minimal submanifolds some known results for minimal submanifolds. Our main tool is a Bochner-type technique involving a formula on the Laplacian of a symmetric function on the Kaehler angles and the Weitzenboeck formula for the Kaehler form of N restricted to M.

math.DG↗
arXiv2000

Minimal submanifolds of Kaehler-Einstein manifolds with equal Kaehler angles

We consider $F: M \to N$ a minimal oriented compact real 2n-submanifold M, immersed into a Kaehler-Einstein manifold N of complex dimension 2n, and scalar curvature R. We assume that $n \geq 2$ and F has equal Kaehler angles. Our main result is to prove that, if n = 2 and $R \neq 0$, then F is either a complex submanifold or a Lagrangian submanifold. We also prove that, if $n \geq 3$ and F has no complex points, then: (A) If R < 0, then F is Lagrangian; (B) If R = 0, the Kaehler angle must be constant. We also study pluriminimal submanifolds with equal Kaehler angles, and prove that, if they are not complex submanifolds, N must be Ricci-flat and there is a natural parallel homothetic isomorphism between TM and the normal bundle.

math.DG↗
arXiv2000

Broadly-Pluriminimal Submanifolds of Kaehler-Einstein Manifolds

We define broadly-pluriminimal immersed 2n-submanifold F: M --> N into a Kaehler-Einstein manifold of complex dimension 2n and scalar curvature R. We prove that, if M is compact, n \geq 2, and R < 0, then: (i) Either F has complex or Lagrangian directions; (ii) If n = 2, M is oriented, and F has no complex directions, then it is a Lagrangian submanifold, generalising the well-known case n = 1 for minimal surfaces due to Wolfson. We also prove that, if F has constant Kaehler angles with no complex directions, and is not Lagrangian, then R = 0 must hold. Our main tool is a formula on the Laplacian of a symmetric function on the Kaehler angles.

math.DG↗