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Curvature
Curves with increasing or decreasing curvature are examined. In particular, the clothoid is discussed. This plays an important role in transportation.
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Heat Flow and Concentration of Measure on Directed Graphs with a Lower Ricci Curvature Bound
In a previous work (Ozawa et al. Calc. Var. Partial Diff. Equ. 59 (4), 39
2020 ), the authors introduced a Lin-Lu-Yau type Ricci curvature for directed... -
Total Gaussian Curvature
If we know a plane curve $$\gamma \colon [a,b]\to \mathbb...
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The metric measure boundary of spaces with Ricci curvature bounded below
We solve a conjecture raised by Kapovitch, Lytchak and Petrunin in [
KLP21 ] by showing that the metric measure boundary is vanishing on any... -
Ollivier Curvature of Random Geometric Graphs Converges to Ricci Curvature of Their Riemannian Manifolds
Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for...
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Boundary behavior of harmonic functions on metric measure spaces with non-negative Ricci curvature
Let ( X, d, μ ) be a metric measure space with non-negative Ricci curvature. This paper is concerned with the boundary behavior of harmonic function on...
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Curvature estimates for a class of Hessian quotient type curvature equations
In this paper, we are concerned with the hypersurface that can be locally represented as a graph and satisfies a class of Hessian quotient type...
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Prescribing the gauss curvature of convex bodies in hyperbolic space
The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to...
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The Uniqueness of Knieper Measure on Non-compact Rank 1 Manifolds of Non-positive Curvature
We study the Knieper measures of the geodesic flows on non-compact rank 1 manifolds of non-positive curvature. We construct the Busemann density on...
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Curvature estimates for semi-convex solutions of Hessian equations in hyperbolic space
In this paper, we establish a curvature estimate for semi-convex solutions of Hessian equations in hyperbolic space. We also obtain a curvature...
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Curvature
This chapter begins by introducing the notion of an isometry (Sect. 5.1). It shows that isometries of embedded manifolds preserve the lengths of...
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Isoperimetric Comparison for Sectional Curvature
We consider in this chapter the validity of the Euclidean isoperimetric inequality in a Cartan–Hadamard manifold, a complete, simply connected...
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Sharp uncertainty principles on metric measure spaces
We prove the rigidity of the Heisenberg–Pauli–Weyl uncertainty principle and the Caffarelli–Kohn–Nirenberg interpolation inequality, on metric...
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Poincaré Inequality and Topological Rigidity of Translators and Self-Expanders for the Mean Curvature Flow
We prove an abstract structure theorem for weighted manifolds supporting a weighted f -Poincaré inequality and whose ends satisfy a suitable...
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Stability of Curvature-Dimension Condition for Negative Dimensions Under Concentration Topology
In this paper, we prove the stability of metric measure spaces satisfying the curvature-dimension condition for negative dimensions under the...
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The mean curvature flow on solvmanifolds
This work is a survey of the most relevant background material to motivate and understand the construction and classification of translating...
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Isothermal Coordinates on Manifolds of Bounded Curvature I
The notion of two-dimensional manifold of bounded curvature was introduced by Alexandrov in [5–7]. Two-dimensional Riemannian manifolds are...
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Curvature bounds on length-minimizing discs
We show that a length-minimizing disk inherits the upper curvature bound of the target. As a consequence we prove that harmonic discs and ruled discs...