We develop a Floer theoretical gluing technique and apply it to deal with the most generic singular ber in the SYZ program, namely the product of a torus with the immersed two-sphere with a single nodal self-intersection. As an application, we construct immersed Lagrangians in Gr(2;Cn) and OG(1;C5) and derive their SYZ mirrors. It recovers the Lie theoretical mirrors constructed by Rietsch. It also gives an eective way to compute stable disks (with non-trivial obstructions) bounded by immersed Lagrangians.

In this paper, we show that any compact K\"ahler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K\"ahler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure $J$ compatible with the symplectic form, there is no non-constant $J$-holomorphic entire curve $f:\C\> X$.

We propose to study the Hessian metric of given functional in the space of probability space embedded with L^2–Wasserstein (optimal transport) metric. We name it transport Hessian metric, which contains and extends the classical L^2–Wasserstein metric. We formulate several dynamical systems associated with transport Hessian metrics. Several connections between transport Hessian metrics and math physics equations are discovered. E.g., the transport Hessian gradient flow, including Newton’s flow, formulates a mean-field kernel Stein variational gradient flow; The transport Hessian Hamiltonian flow of negative Boltzmann-Shannon entropy forms the Shallow water’s equation; The transport Hessian gradient flow of Fisher information forms the heat equation. Several examples and closed-form solutions of finite-dimensional transport Hessian metrics and dynamics are presented.

In this paper, we prove that if a compact Kähler manifold X has a smooth Hermitian metric ω such that (T_X,ω) is uniformly RC-positive, then X is projective and rationally connected. Conversely, we show that, if a projective manifold X is rationally connected, then there exists a uniformly RC-positive complex Finsler metric on T_X .

We generalize the results of Adams–Schoenfeld [2], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each covering a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a uniqueness theorem and demonstrate that many knots cannot possess totally geodesic Seifert surfaces by giving bounds on the width invariant in the presence of such a surface. Finally, we utilize these examples to demonstrate that the Six Theorem is sharp for knot complements in the 3-sphere.