Strong Uniqueness of the Ricci Flow

By Binglong Chen. 2007

455 citations recorded by the source corpus.

In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let $g(t)$ be a smooth complete solution to the Ricci flow on $\mathbb{R}^{3}$, with the canonical Euclidean metric $E$ as initial data, then $g(t)$ is trivial, i.e. $g(t)\equiv E$.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/280eb4d0287c959a6b0ce41d552b838b8ed27e4f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The Ricci flow for simply connected nilmanifolds (2010, background): “The need for this circular argument is due to the fact that the uniqueness of the Ricci flow solution is still an open problem in the noncompact general case (see [18]).”
On Bach-flat gradient shrinking Ricci solitons (2011, background): “Lemma 2.2 is a special case of a more general result of B.-L. Chen [12] which states that R ≥ 0 for any ancient solution to the Ricci ﬂow.”
Rotational symmetry of self-similar solutions to the Ricci flow (2012, background): “It follows from a result of Chen that (M,g) has nonnegative sectional curvature (see [7], Corollary 2.”
Mean Curvature Flow of Mean Convex Hypersurfaces (2013, result): “This is analogous to the theorem that ancient Ricci flows in dimension 3 have nonnegative curvature operator [7, 16].”
On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below (2009, methodology): “In this section, we observe that by a modification of B.-L. Chen’s theorem (Corollary 2.3(i) in [2]), we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds, we follow Yokota’s argument in Proposition A.3 in [10].”
Bach-flat gradient steady Ricci solitons (2011, background): “Lemma 2.2 is a special case of a more general result of B.-L. Chen [10] which states that R ≥ 0 for any ancient solution to the Ricci ﬂow.”
Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, background): “The essential dimension-specific aspects of his argument are, first, that the sectional curvature of a complete steady three-dimensional soliton is (by the Hamilton-Ivey estimate in its local [21] and global [34] forms) necessarily nonnegative and, second, that the asymptotic shrinking soliton obtai”
The conjugate heat equation and Ancient solutions of the Ricci flow (2010, background): “It is well-known that all ancient solutions have nonnegative scalar curvature (for example, see [5]).”
Volume estimate about shrinkers (2011, background): “It is known that the scalar curvature R of the metric g is nonnegative [5].”
On locally conformally flat gradient steady Ricci solitons (2009, unknown): “Chen [7] for 3-dimensional complete ancient solutions.”