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Award ID contains: 2316659

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  1. (, Discrete & Computational Geometry)
    Abstract While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in$$\mathbb {R}^n$$ R n . In doing this they proved a fractional generalization of the Brunn–Minkowski–Lyusternik (BML) inequality in dimension$$n=1$$ n = 1 . In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition$$(\mathcal {G},\beta )$$ ( G , β ) and nonempty sets$$A_1,\dots ,A_m\subseteq \mathbb {R}$$ A 1 , ⋯ , A m ⊆ R , equality holds iff for each$$S\in \mathcal {G}$$ S ∈ G , the set$$\sum _{i\in S}A_i$$ ∑ i ∈ S A i is an interval. In the case of dimension$$n\ge 2$$ n ≥ 2 we will show that equality can hold if and only if the set$$\sum _{i=1}^{m}A_i$$ ∑ i = 1 m A i has measure 0. 
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    Free, publicly-accessible full text available July 1, 2026
  2. ; (, Calculus of Variations and Partial Differential Equations)
    Abstract We prove matrix Li–Yau–Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply these estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature. 
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  3. ; (, The Journal of Geometric Analysis)
    Abstract This article aims to understand the behavior of the curvature operator of the second kind under the Ricci flow in dimension three. First, we express the eigenvalues of the curvature operator of the second kind explicitly in terms of that of the curvature operator (of the first kind). Second, we prove that$$\alpha $$ α -positive/$$\alpha $$ α -nonnegative curvature operator of the second kind is preserved by the Ricci flow in dimension three for all$$\alpha \in [1,5]$$ α ∈ [ 1 , 5 ] . 
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  4. (, Proceedings of the American Mathematical Society)
    Free, publicly-accessible full text available July 29, 2026
  5. ; ; (, The Journal of Geometric Analysis)
    Free, publicly-accessible full text available April 1, 2026
  6. ; ; (, ESAIM: Control, Optimisation and Calculus of Variations)
    Let (Mn,g) be a complete simply connectedn-dimensional Riemannian manifold with curvature bounds Sectg≤ κ for κ ≤ 0 and Ricg≥ (n− 1)KgforK≤ 0. We prove that for any bounded domain Ω ⊂Mnwith diameterdand Lipschitz boundary, if Ω* is a geodesic ball in the simply connected space form with constant sectional curvature κ enclosing the same volume as Ω, then σ1(Ω) ≤Cσ1(Ω*), where σ1(Ω) and σ1(Ω*) denote the first nonzero Steklov eigenvalues of Ω and Ω* respectively, andC=C(n, κ,K,d) is an explicit constant. When κ =K, we haveC= 1 and recover the Brock–Weinstock inequality, asserting that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume, in Euclidean space and the hyperbolic space. 
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    Free, publicly-accessible full text available January 1, 2026
  7. (, Pacific Journal of Mathematics)
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