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Giuseppe Pipoli

Publications and source records attributed to Giuseppe Pipoli.

18 recordsLinked to original sources
arXiv2026

$L^2$-harmonic forms and spinors on stable minimal hypersurfaces

Let $f:N\rightarrow (M,g)$ be an oriented (or spin), complete, stable, minimal, immersed hypersurface. In this paper we establish various vanishing theorems for the space of $L^2$-harmonic forms and spinors (in the spin case) under suitable positive curvature assumptions on the ambient manifold. Our results in the setting of forms extend to higher dimensions and more general ambient Riemannian manifolds previous vanishing theorems due to Tanno \cite{Tanno} and Zhu \cite{Zhu}. In the setting of spin manifolds our results allow to conclude, for instance, that any oriented, complete, stable, minimal, immersed hypersurface of $\mathbb{R}^m$ or $\mathbb{S}^m$ carries no non-trivial $L^2$-harmonic spinors. Finally, analogous results are proved for strongly stable constant mean curvature hypersurfaces.

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arXiv2025

Isoparametric Hypersurfaces in Products of Simply Connected Space Forms

Let $\mathbb Q_{ε_i}^{n_i}$ denote the simply connected space form of dimension $n_i\ge 2$ and constant sectional curvature $ε_i$. We prove that any connected isoparametric hypersurface of $\mathbb Q_{ε_1}^{n_1}\times\mathbb Q_{ε_2}^{n_2}$ has constant angle function. We then use this property to classify the isoparametric and homogeneous hypersurfaces of $\mathbb Q_{ε_1}^{n_1}\times\mathbb Q_{ε_2}^{n_2}$, $|ε_1|+|ε_2|\ne 0$, that satisfy a one-point condition.

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arXiv2025

On the geometry of the asymptotic boundary of translators in $\mathbb H^2\times \mathbb R$

In this work, we study complete properly immersed translators in the product space $\mathbb H^2\times\mathbb R$, focusing on their asymptotic behavior at infinity. We classify the asymptotic boundary components of these translators under suitable continuity assumptions. Specifically, we prove that if a boundary component lies in the vertical asymptotic boundary, it is of the form $\{p\}\times [T,\infty)$ or $\{p\}\times \mathbb R$, while if it lies in the horizontal asymptotic boundary, it is a complete geodesic. Our approach is inspired by earlier work on minimal and constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$, with a key ingredient being the use of symmetric translators as barriers.

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arXiv2025

Translators to Higher Order Mean Curvature Flows in $\mathbb R^n\times\mathbb R$ and $\mathbb H^n\times\mathbb R$

We consider translators to the extrinsic flows in $\mathbb R^n\times\mathbb R$ and $\mathbb H^n\times\mathbb R$ (called $r$-mean curvature flows or $r$-MCF, for short) whose velocity functions are the higher order mean curvatures $H_r.$ We show that there exist rotational bowl-type and catenoid-type translators to $r$-MCF in both $\mathbb R^n\times\mathbb R$ and $\mathbb H^n\times\mathbb R,$ and also that there exist parabolic and hyperbolic catenoid-type translators to $r$-MCF in $\mathbb H^n\times\mathbb R.$ In addition, we show that there exist Grim Reaper-type translators to Gaussian flow ($n$-MCF) in $\mathbb R^n\times\mathbb R$ and $\mathbb H^n\times\mathbb R$. We also establish the uniqueness of all these translators (together with certain cylinders) among those which are invariant by either rotations or translations (Euclidean, parabolic or hyperbolic). We apply this uniqueness result to classify the translators to $r$-MCF in $\mathbb R^n\times\mathbb R$ and $\mathbb H^n\times\mathbb R$ whose $r$-th mean curvature is constant, as well as those which are isoparametric. Our results extend to the context of $r$-MCF in $\mathbb R^n\times\mathbb R$ and $\mathbb H^n\times\mathbb R$ the existence and uniqueness theorems by Altschuler--Wu (of the bowl soliton) and Clutterbuck--Schnürer--Schulze (of the translating catenoids) in Euclidean space.

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arXiv2024

Special Weingarten surfaces with planar convex boundary

We prove a Ros-Rosenberg theorem in the setting of Special Weingarten surfaces. We show that a compact, connected, embedded, Special Weingarten surface in $\mathhb{R}^3$ with planar convex boundary is a topological disk under mild suitable assumptions.

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arXiv2023

On constant higher order mean curvature hypersurfaces in $\mathbb H^n \times \mathbb R$

We classify hypersurfaces with rotational symmetry and positive constant $r$-th mean curvature in $\mathbb H^n \times \mathbb R$. Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also treated. Some of these invariant hypersurfaces are employed as barriers to prove a Ros--Rosenberg type theorem in $\mathbb H^n \times \mathbb R$: we show that compact connected hypersurfaces of constant $r$-th mean curvature embedded in $\mathbb H^n \times [0,\infty)$ with boundary in the slice $\mathbb H^n \times \{0\}$ are topological disks under suitable assumptions.

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arXiv2020

Nonhomogeneous expanding flows in hyperbolic spaces

A recent paper [CGT] studies the evolution of star-shaped mean convex hypersurfaces of the Euclidean space by a class of nonhomogeneous expanding curvature flows. In the present paper we consider the same problem in the real, complex and quaternionic hyperbolic spaces, investigating how the richer geometry of the ambient space affects the evolution. In every case the initial conditions are preserved and the long time existence of the flow is proven. The geometry of the ambient space influences the asymptotic behaviour of the flow: after a suitable rescaling the induced metric converges to a conformal multiple of the standard Riemannian round metric of the sphere if the ambient manifold is the real hyperbolic space, otherwise it converges to a conformal multiple of the standard sub-Riemannian metric on the odd-dimensional sphere. Finally, in every cases, we are able to construct infinitely many examples such that the limit does not have constant scalar curvature.

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arXiv2019

Invariant translators of the Solvable group

We classify the translators to the mean curvature flow in the three-dimensional solvable group $Sol_3$ that are invariant under the action of a one-parameter group of isometries of the ambient space. In particular we show that $Sol_3$ admits graphical translators defined on a half-plane, in contrast with a rigidity result of Shahriyari for translators in the Euclidean space. Moreover we exhibit some non-existence results.

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arXiv2018

Invariant translators of the Heisenberg group

We classify all the translating solitons to the mean curvature flow in the three-dimensional Heisenberg group that are invariant under the action of some one-parameter group of isometries of the ambient manifold. The problem is solved considering any canonical deformation of the standard Riemannian metric of the Heisenberg group. We highlight similarities and differences with the analogous Euclidean translators: we mention in particular that we describe the analogous of the tilted grim reaper cylinders, of the bowl solution and of translating catenoids, but some of them are not convex in contrast with a recent result of Spruck and Xiao in the Euclidean space. Moreover we also prove some negative results. Finally we study the convergence of these surfaces as the ambient metric converges to the standard sub-Riemannian metric on the Heisenberg group.

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arXiv2018

Inverse mean curvature flow in complex hyperbolic space

We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub- Riemannian metric on the sphere. Finally we show that there exists a family of examples such that the Webster curvature of this sub-Riemannian limit is not constant.

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arXiv2017

Inverse mean curvature flow in quaternionic hyperbolic space

In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the quaternionic hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub-Riemannian metric on the sphere defined on a codimension 3 distribution. Finally we show that there exists a family of examples such that the qc-scalar curvature of this sub-Riemannian limit is not constant.

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arXiv2017

Volume preserving non homogeneous mean curvature flow in hyperbolic space

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. The proof is ispired by [10] and is based on the preserving of the convexity by horospheres that allows to bound the inner and outer radii and to give uniform bounds on the curvature by maximum principle arguments. In order to deduce the exponential trend, we study the behaviour of a suitable ratio associated to the hypersurface that converges exponentially in time to the value associated to a geodesic sphere.

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arXiv2016

Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere.

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arXiv2016

Mean curvature flow of pinched submanifolds of $\mathbb{CP}^n$

We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and satisfies a suitable pinching condition on the second fundamental form, then the evolution has two possible behaviors: either the submanifold shrinks to a round point in finite time, or it converges smoothly to a totally geodesic limit in infinite time. The latter behavior is only possible if the dimension is even. These results generalize previous works by Huisken and Baker on the mean curvature flow of submanifolds of the sphere.

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arXiv2015

Mean curvature flow and Riemannian submersions

We give a sufficient condition ensuring that the mean curvature flow commutes with a Riemannian submersion and we use this result to create new examples of evolution by mean curvature flow. In particular we consider evolution of pinched submanifolds of the sphere, of the complex projective space, of the Heisenberg group and the tangent sphere bundle equipped with the Sasaki metric.

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arXiv2013

Lagrangian mean curvature flow of pinched submanifolds of CP^n

We consider the evolution by mean curvature flow of Lagrangian submanifolds of the complex projective space CP^n. We prove that, if the initial value satisfies a suitable pinching condition, then the flow exists for all times and the manifold converges to a totally geodesic submanifold. As a corollary, we obtain that a Lagrangian submanifold satisfying our pinching condition is diffeomorphic to a real projective space.

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