arXiv subjects

Ramiro Lafuente

Publications and source records attributed to Ramiro Lafuente.

5 recordsLinked to original sources

arXiv2024

Ancient Ricci flows of bounded girth

For each $n\ge 3$, we construct a 'pancake-like', $O(2)\times O(n-1)$-invariant ancient Ricci flow with positive curvature operator and bounded "girth", and we determine its asymptotic limits backwards in time. This solution is new even in dimension three. The construction hinges on the Ricci flow invariance of certain conditions on the curvature and its spatial derivatives under this symmetry regime, whose proof does not follow from Hamilton's tensor maximum principle.

math.DG↗

arXiv2016

Optimal curvature estimates for homogeneous Ricci flows

We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension $n$. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on the same space. The above curvature estimates are proved using a gap theorem for Ricci-flatness on homogeneous spaces. The proof of this gap theorem is by contradiction and uses a local $W^{2,p}$ convergence result, which holds without symmetry assumptions.

math.DG↗

arXiv2013

Homogeneous Ricci solitons in low dimensions

In this article we classify expanding homogeneous Ricci solitons up to dimension 5, according to their presentation as homogeneous spaces. We obtain that they are all isometric to solvsolitons, and this in particular implies that the generalized Alekseevskii conjecture holds in these dimensions. In addition, we prove that the conjecture holds in dimension 6 provided the transitive group is not semisimple.

math.DG↗

arXiv2013

Structure of homogeneous Ricci solitons and the Alekseevskii conjecture

We bring new insights into the long-standing Alekseevskii conjecture, namely that any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space, by proving structural results which are actually valid for any homogeneous expanding Ricci soliton, and generalize many well-known results on Einstein solvmanifolds, solvsolitons and nilsolitons. We obtain that any homogeneous expanding Ricci soliton M=G/K is diffeomorphic to a product U/K x N, where U is a maximal reductive Lie subgroup of G and N is the maximal nilpotent normal subgroup of G, such that the metric restricted to N is a nilsoliton. Moreover, strong compatibility conditions between the metric and the action of U on N by conjugation must hold, including a nice formula for the Ricci operator of the metric restricted to U/K. Our main tools come from geometric invariant theory. As an application, we give many Lie theoretical characterizations of algebraic solitons, as well as a proof of the fact that the following a priori much stronger result is actually equivalent to Alekseevskii's conjecture: Any expanding algebraic soliton is diffeomorphic to a Euclidean space.

math.DG↗

arXiv2012

On homogeneous Ricci solitons

We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space of all homogeneous spaces of dimension n with a q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous manifolds. We prove that algebraic solitons (i.e. the Ricci operator is a multiple of the identity plus a derivation) are precisely the fixed points of the system, and that a homogeneous Ricci soliton is isometric to an algebraic soliton if and only if the corresponding bracket flow solution is not chaotic, in the sense that its omega-limit set consists of a single point. We also geometrically characterize algebraic solitons among homogeneous Ricci solitons as those for which the Ricci flow solution is simultaneously diagonalizable.

math.DG↗