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1. Recovery of a time-dependent potential in hyperbolic equations on conformally transversally anisotropic manifolds

   https://doi.org/10.4171/jst/547  

   Liu, Boya; Saksala, Teemu; Yan, Lili (March 2025, Journal of Spectral Theory)

   We study an inverse problem of determining a time-dependent potential appearing in the wave equation on conformally transversally anisotropic manifolds of dimension three or higher. These are compact Riemannian manifolds with boundary that are conformally embedded in a product of the real line and a transversal manifold. Under the assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove the unique determination of time-dependent potentials from the knowledge of a certain partial Cauchy data set. 

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2. Partial data inverse problems for magnetic Schrödinger operators on conformally transversally anisotropic manifolds

   https://doi.org/10.3233/ASY-241909  

   Selim, Salem; Yan, Lili (October 2024, Asymptotic Analysis)

   We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n ⩾ 3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective. 

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3. Inference for Gaussian Processes with Matern Covariogram on Compact Riemannian Manifolds

   Li, D; Tang, W; Banerjee, S (March 2023, Journal of machine learning research)

   Gaussian processes are widely employed as versatile modelling and predictive tools in spa- tial statistics, functional data analysis, computer modelling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on Gaussian processes over Riemannian manifolds in order to develop richer and more flexible inferential frameworks for non-Euclidean data. While numerical approximations through graph representations have been well studied for the Mat´ern covariogram and heat kernel, the behaviour of asymptotic inference on the param- eters of the covariogram has received relatively scant attention. We focus on asymptotic behaviour for Gaussian processes constructed over compact Riemannian manifolds. Build- ing upon a recently introduced Mat´ern covariogram on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Mat´ern Gaussian random measures on compact manifolds to derive the parameter that is identifiable, also known as the microergodic parameter, and formally establish the consistency of the maximum like- lihood estimate and the asymptotic optimality of the best linear unbiased predictor. The circle is studied as a specific example of compact Riemannian manifolds with numerical experiments to illustrate and corroborate the theory 

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4. Classification of asymptotically conical Calabi–Yau manifolds

   https://doi.org/10.1215/00127094-2023-0030  

   Conlon, Ronan J.; Hein, Hans-Joachim (April 2024, Duke Mathematical Journal)

   A Riemannian cone (C,gC) is by definition a warped product C=R+×L with metric gC=dr2⊕r2gL, where (L,gL) is a compact Riemannian manifold without boundary. We say that C is a Calabi-Yau cone if gC is a Ricci-flat Kähler metric and if C admits a gC-parallel holomorphic volume form; this is equivalent to the cross-section (L,gL) being a Sasaki-Einstein manifold. In this paper, we give a complete classification of all smooth complete Calabi-Yau manifolds asymptotic to some given Calabi-Yau cone at a polynomial rate at infinity. As a special case, this includes a proof of Kronheimer's classification of ALE hyper-Kähler 4-manifolds without twistor theory. 

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5. Extremality and rigidity for scalar curvature in dimension four

   https://doi.org/10.1007/s00029-023-00892-5  

   Bettiol, Renato G.; Goodman, McFeely Jackson (February 2024, Selecta Mathematica)

   Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained analyzing sections in the kernel of a twisted Dirac operator constructed from pairs of metrics, and using the Finsler–Thorpe trick for sectional curvature bounds in dimension 4. 

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