Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.0150

Existence and non-existence results for a logistic-type equation on manifolds

Stefano Pigola·Marco Rigoli·Alberto G. Setti
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the steady state solutions of a generalized logistic type equation on a complete Riemannian manifold. We provide sufficient conditions for existence, respectively non-existence of positive solutions, which depend on the relative size of the coefficients and their mutual interaction with the geometry of the manifold, which is mostly taken into account by means of conditions on the volume growth of geodesic balls.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stefano Pigola, Marco Rigoli, Alberto G. Setti. 2007-06-01. Existence and non-existence results for a logistic-type equation on manifolds. https://arxiv.org/abs/0706.0150

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Equivariant analytic torsion for proper actions

We construct an equivariant version of Ray-Singer analytic torsion for proper, isometric actions by locally compact groups on Riemannian manifolds, with compact quotients. We obtain results on convergence, metric independence, vanishing for even-dimensional manifolds, a product formula, and a decomposition of classical Ray-Singer analytic torsion as a sum over conjugacy classes of equivariant torsion on universal covers. We do explicit computations for the circle and the line acting on themselves, and for regular elliptic elements of $\operatorname{SO}_0(3,1)$ acting on $3$-dimensional hyperbolic space. Our constructions and results generalise several earlier constructions of equivariant analytic torsion and their properties, most of which apply to finite or compact groups, or to fundamental groups of compact manifolds acting on their universal covers. These earlier versions of equivariant analytic torsion were associated to finite or compact conjugacy classes; we allow noncompact conjugacy classes, under suitable growth conditions.

math.DG↗
arXiv2026

Contact Blow-up Analysis of the Legendrian Mean Curvature Flow and a Smale-type Theorem for Strict Contactomorphisms of Sasakian Spheres

We study the Legendrian mean curvature flow in $η$-Einstein Sasakian manifolds by parabolic rescalings which preserve the contact structure. In a compact $η$-Einstein Sasakian manifold, regular or not, the contact tangent flows at a Type I singularity are, up to vertical translations of their components, self-similar flows of Legendrian self-shrinkers in the Heisenberg group, and they are unions of Legendrian planes. When the Reeb orbits are not closed, as on the irregular Sasaki--Einstein manifolds, the leaf space of the Reeb foliation is not a Kähler manifold or orbifold, and this result does not follow by projecting the flow to a Lagrangian mean curvature flow; its proof combines local transverse estimates with a Gaussian density localised in a flow box of the Reeb foliation. In a regular manifold a Type I singularity can only form at a point of transverse Gaussian density at least $2$. We apply the analysis to strict contactomorphisms $φ$ of the standard Sasakian sphere $\mathbb S^{2k+1}$, whose Legendrian graphs lie in the $η$-Einstein manifold $(\mathbb S^{2k+1}\times\mathbb S^{2k+1})/\mathbb S^1$. The Jacobian of a Legendrian graph obeys a reaction--diffusion equation whose quadratic term is a sum of squares when $k=1$ and is indefinite when $k\geq2$. For every strict contactomorphism of $\mathbb S^3$, and for every pinched strict contactomorphism of $\mathbb S^{2k+1}$, $k\geq2$, the contact blow-ups of the flow of its Legendrian graph are flat, the flow exists for all time and, up to a rotation along the Reeb flow, converges exponentially to the graph of an element of $U(k+1)$. In particular such a $φ$ is isotopic to $U(k+1)$ through strict contactomorphisms.

math.DG↗
arXiv2026

Uniqueness of Ricci flow with scaling invariant estimates

In this work, we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature. In dimension three, we use it to show that complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique, extending the strong uniqueness Theorem of Chen. This is based on solving Ricci-harmonic map heat flow in unbounded curvature background.

math.DG↗