Topic
Scalar curvature
About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.
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68 citations
TL;DR: This paper showed that a non-flat Ricci shrinker has at most quadratic scalar curvature decay and showed that this result holds for non-compact Ricci solitons.
68 citations
TL;DR: In this paper, it was shown that the Gaussian curvature of a sequence of locally bounded minimal surfaces with locally bounded genus in a homogeneously regular Riemannian 3-manifold is uniformly bounded.
Abstract: Let \mathcal M be the space of properly embedded minimal surfaces in ℝ3 with genus zero and two limit ends, normalized so that every surface M ∊ \mathcal M has horizontal limit tangent plane at infinity and the vertical component of its flux equals one. We prove that if a sequence {M(i)} i ∊ \mathcal M has the horizontal part of the flux bounded from above, then the Gaussian curvature of the sequence is uniformly bounded. This curvature estimate yields compactness results and the techniques in its proof lead to a number of consequences, concerning the geometry of any properly embedded minimal surface in ℝ3 with finite genus, and the possible limits through a blowing-up process on the scale of curvature of a sequence of properly embedded minimal surfaces with locally bounded genus in a homogeneously regular Riemannian 3-manifold.
68 citations
TL;DR: In this article, G 2 -manifolds with a cohomogeneity-one action of a compact Lie group G are studied and the topological types of the manifolds admitting such structures are determined.
68 citations
TL;DR: In this paper, the Lagrangian immersion of a Riemannian manifold Mn to a complex space form of complex dimension n was studied. And the authors constructed and characterized a totally real minimal immersion of S3 in ℂP3 in the complex projective space.
Abstract: In a previous paper, B.-Y. Chen defined a Riemannian invariant δ by subtracting from the scalar curvature at every point of a Riemannian manifold the smallest sectional curvature at that point, and proved, for a submanifold of a real space form, a sharp inequality between δ and the mean curvature function. In this paper, we extend this inequality to totally real submanifolds of a complex space form. As a consequence, we obtain a metric obstruction for a Riemannian manifold Mn to admit a minimal totally real (i.e. Lagrangian) immersion into a complex space form of complex dimension n. Next we investigate three-dimensional submanifolds of the complex projective space ℂP3 which realise the equality in the inequality mentioned above. In particular, we construct and characterise a totally real minimal immersion of S3 in ℂP3.
68 citations