In this paper, we revisit the global well-posedness of the classical viscous surface waves in the absence of surface tension effect with the reference domain being the horizontal infinite slab, for which the first complete proof was given in Guo–Tice [Anal. PDE 6,1429–1533 (2013)] via a hybrid of Eulerian and Lagrangian schemes. The fluid dynamics are governed by the gravity-driven incompressible Navier–Stokes equations. Even though Lagrangian formulation is most natural to study free boundary value problems for incompressible flows, few mathematical works for global existence are based on such an approach in the absence of surface tension effect, due to breakdown of Beale’s transformation. We develop a mathematical approach to establish global well-posedness based on the Lagrangian framework by analyzing suitable “good unknowns” associated with the problem, which requires no nonlinear compatibility conditions on the initial data.
A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space. On the other hand, we provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green’s function and the positive mass theorem play essential roles in certain cases.
In this paper, we show that the generating function for linear Hodge integrals over moduli spaces of stable maps to a nonsingular projective variety X can be connected to the generating function for Gromov–Witten invariants of X by a series of differential operators $\{ L_m \mid m \ge 1 \}$ after a suitable change of variables. These operators satisfy the Virasoro bracket relation and can be seen as a generalization of the Virasoro operators appeared in the Virasoro constraints for Kontsevich–Witten tau-function in the point case. This result is a generalization of the work in Liu and Wang [Commun. Math. Phys. 346(1):143–190, 2016] for the point case which solved a conjecture of Alexandrov.
In this paper, we use the Sobolev type inequality in Wang et al. (Moser–Trudinger inequality for the complex Monge–Ampère equation, arXiv:2003.06056v1 (2020)) to establish the uniform estimate and the Hölder continuity for solutions to the complex Monge–Ampère equation with the right-hand side in $L^p$ for any given $p>1$. Our proof uses various PDE techniques but not the pluripotential theory.