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1. Inverse Problem of Travel Time Difference Functions on a Compact Riemannian Manifold with Boundary

   https://doi.org/10.1007/s12220-018-00111-0  

   de Hoop, Maarten V.; Saksala, Teemu (January 2019, The Journal of Geometric Analysis)

   We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differential structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance map determines the Finsler function in this set but not in its exterior. If the Finsler function is fiberwise real analytic, it is determined uniquely. We also discuss the smoothness of the distance function between interior and boundary points. 

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2. Regularity of viscosity solutions of the σk$$\sigma _k$$‐Loewner–Nirenberg problem

   https://doi.org/10.1112/plms.12536  

   Li, YanYan; Nguyen, Luc; Xiong, Jingang (May 2023, Proceedings of the London Mathematical Society)

   Abstract We study the regularity of the viscosity solution of the ‐Loewner–Nirenberg problem on a bounded smooth domain for . It was known that is locally Lipschitz in . We prove that, with being the distance function to and sufficiently small, is smooth in and the first derivatives of are Hölder continuous in . Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of and its covariant derivatives and vanishes if and only if is smooth in . Using a relation between the Schouten tensor of the ambient manifold and the mean curvature of a submanifold and related tools from geometric measure theory, we further prove that, when contains more than one connected components, is not differentiable in . 

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3. An analogue of polynomially integrable bodies in even-dimensional spaces

   https://doi.org/10.1016/j.jmaa.2023.127071  

   Agranovsky, M; Koldobsky, A; Ryabogin, D; Yaskin, V. (January 2024, Journal of mathematical analysis and applications)

   A bounded domain $$K \subset R^n$$ is called polynomially integrable ifthe $(n-1)$-dimensional volume of the intersection $$K$$ with a hyperplane $$\Pi$$ polynomially depends on the distance from $$\Pi$$ to the origin. It was proved in \cite{KMY} that there are no such domains with smooth boundary if $$n$$ is even, and if $$n$$ is odd then the only polynomially integrable domains with smooth boundary are ellipsoids. In this article, we modify the notion of polynomial integrability for even $$n$$ and consider bodies for which the sectional volume function is a polynomial up to a factor which is the square root of a quadratic polynomial, or, equivalently, the Hilbert transform of this function is a polynomial. We prove that ellipsoids in even dimensions are the only convex infinitely smooth bodies satisfying this property. 

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4. Singularity formation in the harmonic map flow with free boundary

   https://doi.org/10.1353/ajm.2023.a902958  

   Sire, Yannick; Wei, Juncheng; Zheng, Youquan (August 2023, American Journal of Mathematics)

   abstract: In the past years, there has been a new light shed on the harmonic map problem with free boundary in view of its connection with nonlocal equations. Here we fully exploit this link, considering the harmonic map flow with free boundary $$ (0.1)\hskip77pt\cases{u_t=\Delta u& in $$\Bbb{R}^2_+\times (0,T)$$,\cr u(x,0,t)\in\Bbb{S}^1& for all $$(x,0,t)\in\partial\Bbb{R}^2_+\times (0,T)$$,\cr {du\over dy}(x,0,t)\perp T_{u(x,0,t)}\Bbb{S}^1& for all $$(x,0,t)\in\partial\Bbb{R}^2_+\times (0,T)$$,\cr u(\cdot, 0)=u_0& in $$\Bbb{R}^2_+$} $$ for a function $$u:\Bbb{R}^2_+\times [0,T)\to\Bbb{R}^2$$. Here $$u_0 :\Bbb{R}^2_+\to\Bbb{R}^2$$ is a given smooth map and $$\perp$$ stands for orthogonality. We prove the existence of initial data $$u_0$$ such that (0.1) blows up at finite time with a profile being the half-harmonic map. This answers a question raised by Chen and Lin. 

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5. On cusps of caustics by reflection in two dimensional projective Finsler metrics

   https://doi.org/10.2298/TAM250109004T  

   Tabachnikov, Serge (January 2025, Theoretical and Applied Mechanics)

   Given a projective Finsler metric in a convex domain in the projective plane, that is, a metric in which geodesics are straight lines, consider the respective Finsler billiard system. Choose a generic point inside the table and consider the billiard trajectories that start at this point and undergo N reflection off the boundary. The envelope of the resulting 1-parameter family of straight lines is the Nth caustic by reflection. We prove that, for every N, it has at least four cusps, generalizing a similar result for Euclidean metric, obtained recently jointly with G. Bor. 

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