https://neckpinch.com/authors/shrey-aryan/New arXiv papers by Shrey Aryan, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2608.04855arxiv:2608.04855Wed, 05 Aug 2026 00:00:00 GMTShrey Aryan, Tang-Kai Lee — Grayson and Gage proved that an immortal curve shortening flow of simple closed curves on a closed surface converges subsequentially to a closed geodesic, and they asked whether this limiting geodesic is unique. We answer this question negatively by constructing a smooth Riemannian metric on S² and an immortal simple closed curve shortening flow that converges along different sequences of times to every geodesic in a one-parameter family of distinct simple closed geodesics.Curve shorteningStability & uniquenessLong-time existence & convergence https://arxiv.org/abs/2607.27711arxiv:2607.27711Thu, 30 Jul 2026 00:00:00 GMTShrey Aryan, Bang-Xian Han, Zhuo-Nan Zhu — We use inverse mean curvature flow to construct bi-Lipschitz maps that preserve normalized volume and decrease distances. These maps send a round sphere onto any smooth closed strictly convex hypersurface in a sphere and a flat disk onto any smooth strictly convex free-boundary disk in a Euclidean ball. In dimension two, this proves a conjecture of E. Milman for every smooth two-sphere with Gaussian curvature at least one and gives an analogous result for nonnegatively curved disks whose boundary has geodesic curvature one. The spherical result proves the two-dimensional case of the spectral comparison conjectured by Colding and Minicozzi. Counterexamples in dimensions n≥3 show that the restriction to dimension two is sharp. Furthermore, an equivariant extension of this construction yields, for every n≥2, a contracting transport map from the uniform probability measure on a round hemisphere to the uniform probability measure on any closed geodesically convex subset of positive volume. This settles the remaining uniform-target case of a question raised by Beck and Jerison. In dimension two, we also find geometric conditions under which the uniform measure on the hemisphere can be transported by a contracting map to a broad class of nonuniform probability measures supported on domains in a hemisphere.Mean curvature flowOther curvature flowsInverse mean curvature flowSurfacesFree boundary & capillaryEigenvalues & spectrum https://arxiv.org/abs/2605.24705arxiv:2605.24705Sat, 23 May 2026 00:00:00 GMTShrey Aryan — Caffarelli's contraction theorem states that the Brenier optimal transport map from the standard Gaussian measure to a more log-concave probability measure is 1-Lipschitz. Motivated by this theorem, Milman [Mil18] formulated several conjectures for the round sphere and for weighted manifolds satisfying the curvature-dimension condition CD(ρ,∞). Recently, Beck and Jerison [BJ21] raised related questions on the round hemisphere. The existence of a contracting transport map implies a corresponding spectral comparison. In the spherical setting, this comparison was also conjectured by Colding and Minicozzi [CM98] for compact manifolds with Ricci curvature lower bounds. In this work, we construct counterexamples to the corresponding spectral comparisons on spheres and on weighted manifolds satisfying CD(1,∞) in dimensions d≥4, yielding obstructions to contracting transport maps. In dimensions d≥5, the weighted counterexamples can be chosen to satisfy Ric_g≥ 0 and ∇_g² V ≥ g separately. In dimension two, we use inverse mean curvature flow to construct a contracting transport map from the suitably rescaled round sphere to every closed connected Riemannian surface satisfying the same positive Ricci curvature lower bound. This implies the spectral comparison in dimension two. Together with the recent counterexample in dimension three by Lin, Wang, and Xu [LWX26], this settles the spherical spectral comparison in every dimension d≥2. Using the same method, we also construct a contracting transport from the uniform probability measure on a hemisphere onto the normalized uniform measure on any geodesically convex subset of positive volume, thereby answering affirmatively the remaining case of a conjecture by Beck and Jerison [BJ21] following the work of Fathi, Fradelizi, Gozlan, and Zugmeyer [FFGZ26].Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifoldsLimit spaces & RCDEigenvalues & spectrum