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Vicente Miquel

Publications and source records attributed to Vicente Miquel.

16 recordsLinked to original sources
arXiv2023

Geodesic loops on tetrahedra in spaces of constant sectional curvature

Geodesic loops on polyhedra were studied only for Euclidean space and it was known that there are no simple geodesic loops on regular tetrahedra. Here we prove that: 1) On the spherical space, there are no simple geodesic loops on tetrahedra with internal angles $π/3 < α_i<π/2$ or regular tetrahedra with $α_i=π/2$, and there are three simple geodesic loops for each vertex of a tetrahedra with $α_i > π/2$ and the lengths of the edges $alpha_i>π/2$. 2) On the hyperbolic space, for every regular tetrahedron $T$ and every pair of coprime numbers $(p,q)$, there is one simple geodesic loop of $(p,q)$ type through every vertex of $T$.

math.DG↗
arXiv2023

A discrete Blaschke Theorem for convex polygons in $2$-dimensional space forms

Let $M$ be a $2$-space form. Let $P$ be a convex polygon in $M$. For these polygons, we define (and justify) a curvature $κ_i$ at each vertex $A_i$ of the polygon and and prove the following Blaschke's type theorem: If $P$ is a convex plygon in $M$ with curvature at its vertices $κ_i\ge κ_0 >0$, then the circumradius $R$ of $P$ satisfies $ta_λ(R) \le π/(2κ_0)$ and the equality holds if and only if the polygon is a $2$-covered segment.

math.DG↗
arXiv2020

Stability property and Dirichlet problem for translating solitons

In this paper, we prove that the infimum of the mean curvature is zero for a translating solitons of hypersurface in $\re^{n+k}$. We give some conditions under which a complete hypersurface translating soliton is stable. We show that if the norm of its mean curvature is less than one, then the weighted volume may have exponent growth. We also study the Dirichlet problem for graphic translating solitons in higher codimensions.

math.DG↗
arXiv2020

Reilly's type inequality for the Laplacian associated to a density related with shrinkers for MCF

Let $(\bar{M},<,>,e^ψ)$ be a Riemannian manifold with a density, and let $M$ be a closed $n$-dimensional submanifold of $\bar{M}$ with the induced metric and density. We give an upper bound on the first eigenvalue $λ_1$ of the closed eigenvalue problem for $Δ_ψ$ (the Laplacian on $M$ associated to the density) in terms of the average of the norm of the vector ${\vec{H}}_{ψ} + {\bar \nabla}$ with respect to the volume form induced by the density, where ${\vec{H}}_{ψ}$ is the mean curvature of $M$ associated to the density $e^ψ$. When $\bar{M}=\Bbb R^{n+k}$ or $\bar{M}=S^{n+k-1}$, the equality between $λ_1$ and its bound implies that $e^ψ$ is a Gaussian density ($ψ(x) = \frac{C}{2} |x|^2$, $C<0$), and $M$ is a shrinker for the mean curvature flow (MCF) on $\Bbb R^{n+k}$. We prove also that $λ_1 =-C$ on the standard shrinker torus of revolution. Based on this and on the Yau's conjecture on the first eigenvalue of minimal submanifolds of $S^n$, we conjecture that the equality $λ_1=-C$ is true for all the shrinkers of MCF in $\mathbb{R}^{n+k}$.

math.DG↗
arXiv2018

Evolution by mean curvature flow of Lagrangian spherical surfaces in complex Euclidean plane

We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane $\mathbb{C}^2$. In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in $\mathbb{C}^2$ such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the Clifford torus.

math.DG↗
arXiv2016

Type I singularities in the curve shortening flow associated to a density

We define Type I singularities for the mean curvature flow associated to a density $ψ$ ($ψ$MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the $ψ$-curvature due to the density. We describe a family of curves whose evolution under $ψ$MCF (in a Riemannian surface of non-negative curvature with a density which is singular at a geodesic of the surface) produces only type I singularities and study the limits of their blow-ups.

math.DG↗
arXiv2015

The curve shortening problem associated to a density

In $\mathbb{R}^n$ with a density $e^ψ$, we study the mean curvature flow associated to the density ($ψ$-mean curvature flow or $ψ$MCF) of a hypersurface. The main results concern with the description of the evolution under $ψ$MCF of a closed embedded curve in the plane with a radial density, and with a statement of subconvergence to a $ψ$-minimal closed curve in a surface under some general circumstances.

math.DG↗
arXiv2014

Non-preserved curvature conditions under constrained mean curvature flows

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.

math.DG↗
arXiv2011

Bounding the first Dirichlet eigenvalue of a tube around a complex submanifold of $CP^n$ by the degrees of the polynomials defining it

We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model centers of the tube. The bounds are sharp on these models. Moreover, when the models used are $CP^q$ or the complex hyperquadric, these bounds also give gap phenomena and comparison results.

math.DG↗
arXiv2011

Remarks on scalar curvature of Yamabe solitons

In this paper, we consider the scalar curvature of Yamabe solitons. In particular we show that, with natural conditions and non positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that the quadratic decay at infinity of the Ricci curvature of a complete non-compact Yamabe soliton has non-negative scalar curvature. A new proof of Kazdan-Warner condition is also presented.

math.DG↗
arXiv2010

Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants

In a rotationally symmetric space $\oM$ around an axis A (whose precise definition includes all real space forms), we consider a domain $G$ limited by two equidistant hypersurfaces orthogonal to A. Let $M \subset \oM$ be a revolution hypersurface generated by a graph over A, with boundary in $\partial G$ and orthogonal to it. We study the evolution $M_t$ of $M$ under the volume-preserving mean curvature flow requiring that the boundary of $M_t$ rests on $\partial G$ and keeps orthogonal to it. We prove that: a) the generating curve of $M_t$ remains a graph; b) the flow exists while $M_t$ does not touch the axis of rotation; c) under a suitable hypothesis relating the enclosed volume and the area of $M$, the flow is defined for every $t\in [0,\infty[$ and a sequence of hypersurfaces $M_{t_n}$ converges to a revolution hypersurface of constant mean curvature. Some key points are: i) the results are true even for ambient spaces with positive curvature, ii) the averaged mean curvature does not need to be positive and iii) for the proof it is necessary to carry out a detailed study of the boundary conditions.

math.DG↗
arXiv2009

Gaussian Mean curvature flow

We consider the evolution of a $n$-dimensional convex hypersurface in the euclidean space under mean curvature flow with densities $e^{\varepsilon \frac12 nμ^2 |x|^2}$, $\varepsilon =\pm 1$, and completely determine it depending on the relation between $μ$ and the upper or lower bound of the normal curvatures of the evolving hypersurface at time 0.

math.DG↗
arXiv2009

Mean curvature flow of graphs in warped products

Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $φ$ be a positive smooth function on $M$. In the warped product $M\times_φ\mathbb R$, we study the flow by the mean curvature of a locally Lipschitz continuous graph on $M$ and prove that the flow exists for all time and that the evolving hypersurface is $C^\infty$ for $t>0$ and is a graph for all $t$. Moreover, under certain conditions, the flow has a well defined limit.

math.DG↗
arXiv2008

Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t, the evolving hypersurface M_t meets such tgh ortogonally, we prove that: a) the flow exists while M_t does not touch the axis of rotation; b) throughout the time interval of existence, b1) the generating curve of M_t remains a graph, and b2) the averaged mean curvature is double side bounded by positive constants; c) the singularity set (if non-empty) is finite and discrete along the axis; d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature.

math.DG↗
arXiv2006

Volume preserving mean curvature flow in the Hyperbolic space

We prove: "If $M$ is a compact hypersurface of the hyperbolic space, convex by horospheres and evolving by the volume preserving mean curvature flow, then it flows for all time, convexity by horospheres is preserved and the flow converges, exponentially, to a geodesic sphere". In addition, we show that the same conclusions about long time existence and convergence hold if $M$ is not convex by horospheres but it is close enough to a geodesic sphere.

math.DG↗