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1. Uniqueness of Blowups for Forced Mean Curvature Flow

   https://doi.org/10.1093/imrn/rnaf023  

   Hirsch, Sven; Zhu, Jonathan J (February 2025, International Mathematics Research Notices)

   Abstract We prove uniqueness of tangent cones for forced mean curvature flow, at both closed self-shrinkers and round cylindrical self-shrinkers, in any codimension. The corresponding results for mean curvature flow in Euclidean space were proven by Schulze and Colding–Minicozzi, respectively. We adapt their methods to handle the presence of the forcing term, which vanishes in the blow-up limit but complicates the analysis along the rescaled flow. Our results naturally include the case of mean curvature flows in Riemannian manifolds. 

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2. Rigidity properties of Colding–Minicozzi entropies

   https://doi.org/10.1515/ans-2022-0082  

   Bernstein, Jacob (February 2024, Advanced Nonlinear Studies)

   Abstract We show certain rigidity for minimizers of generalized Colding–Minicozzi entropies. The proofs are elementary and work even in situations where the generalized entropies are not monotone along mean curvature flow. 

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3. Colding–Minicozzi entropies in Cartan–Hadamard manifolds

   https://doi.org/10.1515/crelle-2025-0032  

   Bernstein, Jacob; Bhattacharya, Arunima (May 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We introduce a family of functionals defined on the set of submanifolds of Cartan–Hadamard manifolds which generalize the Colding–Minicozzi entropy of submanifolds of Euclidean space.We show these functionals are monotone under mean curvature flow under natural conditions.As a consequence, we obtain sharp lower bounds on these entropies for certain closed hypersurfaces and observe a novel rigidity phenomenon. 

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4. Half-Space Intersection Properties for Minimal Hypersurfaces

   https://doi.org/10.1007/s12220-025-02006-3  

   Naff, Keaton; Zhu, Jonathan J (June 2025, The Journal of Geometric Analysis)

   We prove ``half-space" intersection properties in three settings: the hemisphere, half-geodesic balls in space forms, and certain subsets of Gaussian space. For instance, any two embedded minimal hypersurfaces in the sphere must intersect in every closed hemisphere. Two approaches are developed: one using classifications of stable minimal hypersurfaces, and the second using conformal change and comparison geometry for $$\alpha$$-Bakry-\'{E}mery-Ricci curvature. Our methods yield the analogous intersection properties for free boundary minimal hypersurfaces in space form balls, even when the interior or boundary curvature may be negative. Finally, Colding and Minicozzi recently showed that any two embedded shrinkers of dimension $$n$$ must intersect in a large enough Euclidean ball of radius $R(n)$. We show that $$R(n) \leq 2 \sqrt{n}$$. 

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5. Electronic absorption spectra from off-diagonal quantum master equations

   https://doi.org/10.1063/5.0106888  

   Lai, Yifan; Geva, Eitan (September 2022, The Journal of Chemical Physics)

   Quantum master equations (QMEs) provide a general framework for describing electronic dynamics within a complex molecular system. Off-diagonal QMEs (OD-QMEs) correspond to a family of QMEs that describe the electronic dynamics in the interaction picture based on treating the off-diagonal coupling terms between electronic states as a small perturbation within the framework of second-order perturbation theory. The fact that OD-QMEs are given in terms of the interaction picture makes it non-trivial to obtain Schrödinger picture electronic coherences from them. A key experimental quantity that relies on the ability to obtain accurate Schrödinger picture electronic coherences is the absorption spectrum. In this paper, we propose using a recently introduced procedure for extracting Schrödinger picture electronic coherences from interaction picture inputs to calculate electronic absorption spectra from the electronic dynamics generated by OD-QMEs. The accuracy of the absorption spectra obtained this way is studied in the context of a biexciton benchmark model, by comparing spectra calculated based on time-local and time-nonlocal OD-QMEs to spectra calculated based on a Redfield-type QME and the non-perturbative and quantum-mechanically exact hierarchical equations of motion method. 

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