Pubblications

8. Invariant translators of the Solvable group, preprint

We classify the translators to the mean curvature flow in the three-dimensional solvable group Sol3 that are invariant under the action of a one-parameter group of isometries of the ambient space. In particular we show that Sol3 admits graphical translators defined on a half-plane, in contrast with a rigidity result of Wang for translators in the Euclidean space. Moreover we exhibit some non-existence results.

arXiv link

7. Invariant translators of the Heisenberg group, preprint.

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. We do the classification for 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 non-existence results. Finally we study the convergence of these surfaces as the ambient metric converges to the standard sub-Riemannian metric on the Heisenberg group.

arXiv link

6. Inverse mean curvature flow in quaternionic hyperbolic space, Rendiconti Lincei Matematica e Applicazioni 29 (1), 2018, 153 – 171.

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.

Link to the article arXiv link

5. Volume preserving non homogeneous mean curvature flow in hyperbolic space, with Maria Chiara Bertini, Differential Geometry and its applications (54), 2017, 448– 463.

We study a volume/area preserving curvature flow of h-convex hypersurfaces 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 [7] and is based on the preserving of the h-convexity 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.

Link to the article arXiv link

4. Inverse mean curvature flow in complex hyperbolic space, to appear in Annales scientifiques de l'ENS.

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.

arXiv link

3. Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes, with Carlo Sinestrari, Annals of Global Analysis and Geometry 51 (2) 2017, pp 179 - 188.

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.

Link to the article arXiv link

2. Mean curvature flow and Riemannian submersions, Geometriae Dedicata 184 (1), 2016, pp 67 – 81.

We prove that a sufficient condition ensuring that the mean curvature flow commutes with a Riemannian submersion is that the submersion has minimal fibers. We then lift some results taken from the literature (i.e., Andrews and Baker in J Differ Geom 85:357–395, 2010; Baker in The mean curvature flow of submanifolds of high codimension, 2011; Huisken in J Differ Geom 20:237–266, 1984; Math Z 195:205–219, 1987; Pipoli and Sinestrari in Mean curvature flow of pinched submanifolds of $\mathbb{CP}^n$) to create new examples of evolution by mean curvature flow. In particular we consider the evolution of pinched submanifolds of the sphere, of the complex projective space, of the Heisenberg group and of the tangent sphere bundle equipped with the Sasaki metric.

Link to the article

1. Mean curvature flow of pinched submanifolds of $\mathbb{CP}^n$, with Carlo Sinestrari, Communications in Analysis and Geometry 25 (4), 2017, pp 799 - 846.

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.

Link to the article arXiv link