Some old and new results about rigidity of critical metric

Abstract

We present a new proof of a recent ✏-regularity of G. Tian and J.Viaclovsky. Our idea also provides a new proof of a classical result of M. Anderson about volume rigidity of Einstein manifolds. Eventually, we also obtain new rigidity results for critical metrics.

The partial 𝐶⁰-estimate along the continuity method

“…The analogous result for conical Kähler-Einstein metrics, [13, Proposition 3], is straightforward because I(B) is bounded away from 1 by a definite amount for balls centered at a conical singularity. The proof of the result is reminiscent of arguments in Carron [6].…”

Quantitative $\mathsf{C}^1$-estimates on Manifolds

“…The idea of using a lifting procedure in order to avoid assumptions on the injectivity radius can be traced back to a paper of J. Cheeger and M. Gromov [3], where it has been used to smoothing Riemannian distance functions. The use we make of this idea in the present paper inspires to the recent [2], where applications to critical metrics, ε-regularity results and geometric rigidity are presented. The authors would like to thank G. Carron for having pointed out this reference.…”

Some rigidity results for noncompact gradient steady Ricci solitons and Ricci-flat manifolds

“…We would like to remark that since |∇f | is bounded, the proof in [5] with little modification shows that gradient steady Ricci solitons are (Γ, k) regular in the definition of Carron (see section 4). Therefore, some results in [5] which only depend on the local regularity property still hold for gradient steady Ricci solitons, these results imply that the curvature cannot decay too fast locally.…”

“…With the help of Lemma 4.3, Theorem 4.1 and 4.2 directly yield the following rigidity results: Theorem 4.4. Let (M, g) be a complete Ricci-flat Riemannian manifold with To prove Theorem 1.3, we need to recall the regularity estimates in [5], where Carron defined the following: |∇ j Rm| ≤ Γ δ j r j+2 , j = 1, 2, ..., k. Proof. This lemma is proved in the more general setting of critical metrics in [5].…”

“…Let (M, g) be a complete Ricci-flat Riemannian manifold with To prove Theorem 1.3, we need to recall the regularity estimates in [5], where Carron defined the following: |∇ j Rm| ≤ Γ δ j r j+2 , j = 1, 2, ..., k. Proof. This lemma is proved in the more general setting of critical metrics in [5]. In the special case of Einstein metrics, it can be seen as an elliptic version of Shi's estimate [20].…”