International Journal of Mathematics

In this paper, we study the interesting open problem of classifying the minimal Lagrangian submanifolds of dimension $n$ in complex space forms with semi-parallel second fundamental form. First, we completely solve the problem in cases $n=2,3,4$. Second, supposing further that the scalar curvature is constant for $n\ge 5$, we also give an answer to the problem by applying the classification theorem of [F. Dillen, H. Li, L. Vrancken and X. Wang, Lagrangian submanifolds in complex projective space with parallel second fundamental form, Pacific J. Math. 255 (2012) 79–115]. Finally, for such Lagrangian submanifolds in the above problem with $n\ge 3$, we establish an inequality in terms of the traceless Ricci tensor, the squared norm of the second fundamental form and the scalar curvature. Moreover, this inequality is optimal in the sense that all the submanifolds attaining the equality are completely determined.

The goal of this work is to study geometric properties of geometrically irreducible subschemes on degenerations of Fano varieties (more generally, of separably rationally connected varieties). It is known that these geometrically irreducible subschemes exist when the ground field has characteristic zero or contains an algebraically closed subfield. We show that the dimension of this geometrically irreducible subscheme has a lower bound by the Fano index of the generic fiber.

Parabolic Finslerian Allen–Cahn equation is a quasilinear parabolic equation on Finsler metric measure spaces. In this paper, we establish a Harnack inequality for solutions to this equation on compact Finsler metric measure spaces using geometric methods. Moreover, we give a degree 4 polynomial gradient estimate of the traveling wave solution on flat-curved spaces.

In this paper, we investigate the existence, uniqueness, and asymptotic behaviors of mild solutions of parabolic evolution equations on complex plane, in which the diffusion operator has the form ${\overline{\square }}_{\phi }=\overline{D}{\overline{D}}^{\ast }$, where $\overline{D}f=\bar{\partial }f+{\phi }_{\bar{z}}f$, the function $\phi$ is smooth and subharmonic on $ℂ$, and ${\overline{D}}^{\ast }$ is the formal adjoint of $\overline{D}$. Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich [Heat equations in $ℝ\times ℂ$, J. Funct. Anal. 240(1) (2006) 1–35] and a fixed point argument.

Let $f:S\to C$ be a surjective morphism with connected fibers from a smooth complex projective surface $S$ to a smooth complex projective curve $C$ with general fiber $F$. In this paper, we develop a more general version of the slope inequality for data $(D,{ℱ})$, where $D$ is an arbitrary relatively effective divisor on $S$ and ${ℱ}$ is a locally free sub-sheaf of $f_{\ast }{{𝒪}}_S(D)$. We see how the speciality of $D$, restricted to the general fiber, plays a role in the results. Moreover, we compute some natural examples and provide applications.

A central problem in low-dimensional topology asks which homology three-spheres bound contractible four-manifolds or homology four-balls. In this paper, we address this question for plumbed three-manifolds and we present two new infinite families. We consider most of the classical examples from around the 1980s by reproving that they all bound Mazur manifolds. We also show that several well-known families bound possibly different types of four-manifolds, called Poénaru homology four-balls. To unify classical and new results in a fairly simple way, we modify Mazur’s argument and work with Poénaru manifolds.