Riemannian Geometry: Isoperimetric Inequalities (Constant Curvature)

By I. Chavel. 2006

793 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/f71fff24e24653bd73735ebf3a41a95eb92eaa7a

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In Applied Mathematics

Citations in the corpus

Prekopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport (2006, neutral): “Chavel[12]).”
Conformal properties in classical minimal surface theory (2004, background): “The following statement of the coarea formula appears in [9], see [33] for a more general version.”
A Universal Volume Comparison Theorem for Finsler Manifolds and Related Results (2013, background): “4 and the standard argument (see [13] or [39]), one can show that Q(r) := μ(B+(x, r)) rn is monotonically decreasing in r.”
Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry (2008, background): “In fact for any two dimensional connected and simply connected domain with Gauss-curvature K everywhere less or equal to −κ, Yau [44, 45] has shown that L(Σ) > √ κA(Σ).”
Geometric constraints for shape and topology optimization in architectural design (2017, methodology): “4), we apply the coarea formula to the mapping p∂Ω : Ω \ ΣM → ∂Ω, as a curvilinear version of the classical Fubini theorem (see [20], or [3] in the same context):”
Riemannian Manifold Hamiltonian Monte Carlo (2009, methodology): “…as the tim evolution, with respect to a fictitious time τ , of the physical system as given by Hamilton’s equations, dθ dτ = ∂H ∂p = M−1p dp dτ = −∂H ∂θ = ∇θL(θ) (2) Riemannian Manifold Hamiltonian Monte Carlo 3 These deterministic equations can be exploited in defining aproposal process…”
A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions (2016, neutral): “ −Γ ϕ ϕϕ ∂ ∂ϕ −Γ θ ϕϕ ∂ ∂θ] # = " ∂2 ∂θ2 1 sinθ[ ∂2 ∂θ∂ϕ − cosθ ∂ ∂ϕ] 1 sinθ[ ∂2 ∂θ∂ϕ − cosθ sinθ ∂ ∂ϕ] 1 sin2θ[ ∂2 ∂ϕ2 +sin θcos ∂ ∂θ] # , where Γc ab are the usual Christoﬀel symbols, see e.g. [9] Section I.1, from which we can compute the Levi-Civita connection: ∇~e a ~eb = Γ θ ab~eθ +Γ ϕ ab”
Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries (2011, methodology): “In this framework, in fact, besides the already mentioned Li-Yau gradient estimates, with the ensuing scale invariant Harnack inequality and the Liouville theorem of Yau, see [55], we also obtain an elementary proof of the fundamental monotonicity of Perelman’s entropy for the heat equation, see [44”
Non-parametric regression estimation on closed Riemannian manifolds (2006, background): “For materials on differential and Riemannian geometry, we refer the reader to [5, 21, 13, 30].”
Prescribing symmetric functions of the eigenvalues of the Ricci tensor (2004, neutral): “By the standard version of Bishop’s theorem for smooth metrics, it follows that gw is isometric to the Euclidean metric (see [ Cha93 , Theorem 3.9]).”