This volume contains the proceedings of the AMS Special Session on Recent Advanced in Differential Geometry, held virtually on April 1-2, 2023, and the Tenth Congress of Romanian Mathematicians Special Session on Recent Advances in the Geometry of Submanifolds, held on June 30-July 5, 2023, in Pitesti, Romania. In the last two decades, various techniques produced new results in differential geometry and many of them focused on the geometry of submanifolds, where one of the most important questions, the Willmore conjecture, was solved in 2012 by A. Neves and F. Coda Marques. This invites a reflection on the new trends and research directions that contemporary differential geometry might and will take in its natural development. Some of the most important advances have been obtained by using techniques in geometric analysis, while others have been using more classical methods, which are still of interest.
Motivated by these new developments, the present volume intends to bring together various viewpoints on the recent study of representing the fundamental idea of space in differential geometry. Of particular interest are the results focused on minimal submanifolds and their generalizations and the related concept of harmonic mappings and their generalizations. In preparing this volume, the editors were particularly interested in the study of curvature functionals in various contexts, from comparison geometry to new curvature invariants, relations between curvature and topology, as well as other related topics.
Articles Rares Ambrosie, Harmonicity and biharmonicity of quadratic maps between spheres Mateo Anarella, A survey on submanifolds of nearly Kahler spaces Yusei Aoki and Toshiaki Adachi, Expressions of circles in a complex hyperbolic space by trajectories on tubes around complex hyperplanes Volker Branding, On conservation laws for polyharmonic maps Simona Decu and Gabriel-Eduard Vilcu, Chen inequality for statistical submanifolds in Kenmotsu statistical manifolds of constant $\phi $-sectional curvature Marie D'haene, Thurston geometries in dimension four from a Riemannian perspective Amalendu Ghosh, Ramesh Sharma and Rahul Poddar, Recent results in Ricci and Yamabe solitons in contact geometry Hsiao-Fan Liu, Understanding of constant $p$-mean curvature hypersurfaces in the Heisenberg groups Rafael Lopez, Open problems on compact constant mean curvature surfaces with boundary Andreas Malmendier and Michael T. Schultz, On holomorphic conformal structures associated with lattice polarized K3 surfaces S. Montaldo and A. Ratto, Triharmonic curves in the $3$-dimensional Sol space Ana Irina Nistor, A note on Weingarten surfaces foliated by helices Alvaro Pampano, Characterizations of rotational biconservative hypersurfaces Alexander Pigazzini, Luca Lussardi, Magdalena Toda and Andrew DeBenedictis, Einstein warped-product manifolds and the screened Poisson equation
Bogdan D. Suceava, California State University, Fullerton, CA, Adara M. Blaga, West University of Timisoara, Romania, Marian Ioan Munteanu, Alexandru Ioan Cuza University, Iasi, Romania, Cezar Oniciuc, Alexandru Ioan Cuza University, Iasi, Romania, Shoo Seto, California State University, Fullerton, CA, Gabriel-Eduard Vilcu, Polytechnic University Bucharest, Romania, and Lihan Wang, California State University, Long Beach, CA.
