Gaussian densities and stability for some Ricci solitons

Abstract

In this announcement, we exhibit the second variation of Perelman's $\lambda$ and $\nu$ functionals for the Ricci flow, and investigate the linear stability of examples. We also define the "central density" of a shrinking Ricci soliton and compute its values for certain examples in dimension 4. Using these tools, one can sometimes predict or limit the formation of singularities in the Ricci flow. In particular, we show that certain Einstein manifolds are unstable for the Ricci flow in the sense that generic perturbations acquire higher entropy and thus can never return near the original metric.

15 pages

Cited by in corpus (42)

• Recent Progress on Ricci Solitons
• On Type I Singularities in Ricci flow
• Stability and instability of Ricci solitons
• Dynamical stability and instability of Ricci-flat metrics
• Stability of Einstein metrics under Ricci flow
• Convergence of a Kähler-Ricci flow
• Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures on Lie Algebroids
• Perelman's W-functional and stability of Kähler-Ricci flow
• Sharp logarithmic Sobolev inequalities on gradient solitons and applications
• On the uniqueness of asymptotic limits of the Ricci flow
• Perelman's reduced volume and a gap theorem for the Ricci flow
• A renormalized Perelman-functional and a lower bound for the ADM-mass
• Minimal surfaces and mean curvature flow
• Constructing Kähler-Ricci solitons from Sasaki-Einstein manifolds
• Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations
• Ricci Flow Approach to The Cosmological Constant Problem
• Numerical Kaehler-Ricci soliton on the second del Pezzo
• Local Conformal Instability and Local Non-Collapsing in the Ricci flow of Quantum Spacetime
• An equation linking $\mathcal{W}$-entropy with reduced volume
• Rigidity and infinitesimal deformability of Ricci solitons
• Lagrangian F-stability of closed Lagrangian self-shrinkers
• Perelman's $W$-functional on manifolds with conical singularities
• Boltzmann's Entropy and Kähler-Ricci Solitons
• Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three
• Perelman's Entropies for Manifolds with conical Singularities
• The canonical Einstein metric on $G_2$ is dynamically unstable under the Ricci flow
• On second variation of Perelman's Ricci shrinker entropy
• Dynamic instability of ${\mathbb{CP}}^N$ under Ricci flow
• Linear Instability of Sasaki Einstein and nearly parallel ${\mathrm{G}}_2$ manifolds
• Convergence of compact Ricci solitons
• A compactness result for Fano manifolds and Kähler Ricci flows
• A gap theorem on complete shrinking gradient Ricci solitons
• Commutator formulas for gradient Ricci shrinker and their application to linear stability
• Kähler-Ricci flow for deformed complex structures
• Linear stability of Perelman's $\nu$-entropy of standard Einstein manifolds
• On the linear stability of nearly-Kähler $6$-manifolds
• Perelman's entropy for some families of canonical metrics
• Local monotonicity and mean value formulas for evolving Riemannian manifolds
• Mean value theorems on manifolds
• The stability inequality for Ricci-flat cones
• The second variation of the Ricci expander entropy
• Stabilities of homothetically shrinking Yang-Mills solitons