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1. Uniqueness of convex ancient solutions to mean curvature flow in R^3.

   https://doi.org/10.1007/s00222-019-00859-4  

   Brendle, Simon; Choi, Kyeongsu (July 2019, Inventiones mathematicae)

   A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension 3 which have positive sectional curvature and are κ-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in R^3, and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in R^3 which is strictly convex and noncollapsed. 

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2. Control of eigenfunctions on surfaces of variable curvature

   https://doi.org/10.1090/jams/979  

   Dyatlov, Semyon; Jin, Long; Nonnenmacher, Stéphane (April 2022, Journal of the American Mathematical Society)

   We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schrödinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works (see Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339] and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794]), which considered the setting of surfaces of constant negative curvature. The proofs use the strategy of Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339 and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794] and rely on the fractal uncertainty principle of Jean Bourgain and Semyon Dyatlov [Ann. of Math. (2) 187 (2018), pp. 825–867]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used by Semyon Dyatlov and Joshua Zahl [Geom. Funct. Anal. 26 (2016), pp. 1011–1094]. Instead, our argument uses Egorov’s theorem up to local Ehrenfest time and the hyperbolic parametrix of Stéphane Nonnenmacher and Maciej Zworski [Acta Math. 203 (2009), pp. 149–233], together with the C 1 + C^{1+} regularity of the stable/unstable foliations. 

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3. The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces

   Gong, Sherry; Wu, Jianchao; Yu, Guoliang (August 2024, Geometric and functional analysis)

   We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard spaces include Hilbert spaces, certain simply connected and non-positively curved Riemannian-Hilbertian manifolds and infinite-dimensional symmetric spaces. Thus our main theorem can be considered as an infinite-dimensional analogue of Kasparov’s theorem on the Novikov conjecture for groups acting properly and isometrically on complete, simply connected and non-positively curved manifolds. As a consequence, we show that the Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This result is inspired by Connes’ theorem that the Novikov conjecture holds for higher signatures associated to the Gelfand-Fuchs classes of groups of diffeormorphisms. 

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4. Symmetry of hypersurfaces and the Hopf Lemma

   https://doi.org/10.3934/mine.2023084  

   Li, YanYan (January 2023, Mathematics in Engineering)

   A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed. 

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5. Endperiodic automorphisms of surfaces and foliations

   https://doi.org/10.1017/etds.2019.56  

   CANTWELL, JOHN; CONLON, LAWRENCE; FENLEY, SERGIO R. (January 2021, Ergodic Theory and Dynamical Systems)

   We extend the unpublished work of Handel and Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel–Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic laminations, show that the geodesic laminations satisfy the axioms, and prove that pseudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the ‘transfer theorem’ for foliations of 3-manifolds, namely that, if two depth-one foliations $${\mathcal{F}}$$ and $${\mathcal{F}}^{\prime }$$ are transverse to a common one-dimensional foliation $${\mathcal{L}}$$ whose monodromy on the non-compact leaves of $${\mathcal{F}}$$ exhibits the nice dynamics of Handel–Miller theory, then $${\mathcal{L}}$$ also induces monodromy on the non-compact leaves of $${\mathcal{F}}^{\prime }$$ exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends. 

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