Rigidity of gradient Einstein shrinkers

In this paper, we consider a perturbation of the Ricci solitons equation proposed in [J.-P. Bourguignon, Ricci curvature and Einstein metrics, in Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, Vol. 838 (Springer, Berlin, 1981), pp. 42–63] and studied in [H.-D. Cao, Geometry of Ricci solitons, Chinese Ann. Math. Ser. B27(2) (2006) 121–142] and we classify noncompact gradient shrinkers with bounded non-negative sectional curvature.

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Homogeneous Einstein Finsler Metrics on -dimensional Spheres

Canadian Mathematical Bulletin ◽

10.4153/s0008439518000139 ◽

2018 ◽

Vol 62 (3) ◽

pp. 509-523

Author(s):

Libing Huang ◽

Xiaohuan Mo

Keyword(s):

Sectional Curvature ◽

Ricci Curvature ◽

Einstein Metrics ◽

Constant Sectional Curvature ◽

Einstein Metric ◽

Finsler Metrics ◽

Dimensional Sphere ◽

Order Ordinary Differential Equation ◽

Canonical Metric ◽

Homogeneous Einstein Metrics

AbstractIn this paper, we study a class of homogeneous Finsler metrics of vanishing $S$-curvature on a $(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature $1$ in this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature $1$ and vanishing $S$-curvature. They contain the canonical metric on $S^{4n+3}$ of constant sectional curvature $1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.

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Global Differential Geometry and Global Analysis 1984

10.1007/bfb0075080 ◽

1985 ◽

Keyword(s):

Differential Geometry ◽

Global Analysis ◽

Global Differential Geometry

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Global Differential Geometry and Global Analysis

10.1007/bfb0083621 ◽

1991 ◽

Keyword(s):

Differential Geometry ◽

Global Analysis ◽

Global Differential Geometry

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ON LOCALLY CONFORMALLY FLAT GRADIENT SHRINKING RICCI SOLITONS

Communications in Contemporary Mathematics ◽

10.1142/s0219199711004191 ◽

2011 ◽

Vol 13 (02) ◽

pp. 269-282 ◽

Cited By ~ 22

Author(s):

XIAODONG CAO ◽

BIAO WANG ◽

ZHOU ZHANG

Keyword(s):

Sectional Curvature ◽

Ricci Curvature ◽

Exponential Growth ◽

Integral Identity ◽

Ricci Solitons ◽

Riemannian Curvature ◽

Conformally Flat ◽

Gradient Ricci Solitons ◽

Riemannian Curvature Tensor ◽

Locally Conformally Flat

In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below.

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