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Creators/Authors contains: "Zhu, Jonathan J"

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  1. ; (, 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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    Free, publicly-accessible full text available June 1, 2026
  2. ; (, 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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    Free, publicly-accessible full text available February 13, 2026
  3. (, International Mathematics Research Notices)
    Abstract We prove Łojasiewicz inequalities for round cylinders and cylinders over Abresch–Langer curves, using perturbative analysis of a quantity introduced by Colding–Minicozzi. A feature is that this auxiliary quantity allows us to work essentially at 1st order. This new method interpolates between the higher-order perturbative analysis used by the author for certain shrinking cylinders and the differential geometric method used by Colding–Minicozzi for the round case. 
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  4. ; ; ; (, Communications on Pure and Applied Mathematics)
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  5. ; (, Cambridge journal of mathematics)
    We prove that, for a generic set of smooth prescription functions h on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature h. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersurface of multiplicity one. More precisely, we show that our previous min-max theory, developed for constant mean curvature hypersurfaces, can be extended to construct min-max prescribed mean curvature hypersurfaces for certain classes of prescription function, including a generic set of smooth functions, and all nonzero analytic functions. In particular we do not need to assume that h has a sign. 
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    Accepted Manuscript Full Text Available
  6. ; (, Advances in Mathematics)
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