Related papers: Solvability of a class of fully nonlinear elliptic…
Motivated from mathematical aspects of the superstring theory, we introduce a new equation on a balanced, hermitian manifold, with zero first Chern class. Solving the equation, one will obtain, in each Bott--Chern cohomology class, a…
We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.
This paper pursues the study of the Calabi-Yau equation on certain symplectic non-Kaehler 4-manifolds, building on a key example of Tosatti-Weinkove in which more general theory had proved less effective. Symplectic 4-manifolds admitting a…
In this paper, we study the form type Calabi-Yau equation. We define the astheno-Ricci curvature and prove that there exists a solution for the form type Calabi-Yau equation if the astheno-Ricci curvature is non-positive.
We study the solvability in the whole Euclidean space of coercive quasi-linear and fully nonlinear elliptic equations modeled on $\Delta u\pm g(|\nabla u|)= f(u)$, $u\ge0$, where $f$ and $g$ are increasing continuous functions. We give…
In this paper, we consider a class of fully nonlinear equations on closed smooth Riemannian manifolds, which can be viewed as an extension of $\sigma_k$ Yamabe equation. Moreover, we prove local gradient and second derivative estimates for…
In this paper we establish a necessary and sufficient condition for solving a general class of fully nonlinear elliptic equations on compact almost Hermitian manifolds, extending a recent work of Guo-Song. As applications, we determine the…
We study a class of Calabi-Yau varieties that can be represented as a non-singular model of a double covering of $\mathbb P^3$ branched along certain octic surfaces. We compute Euler numbers of all constructed examples and describe their…
We obtain existence results for a class of fully nonlinear Yamabe-type problems on non-compact manifolds, addressing both the so-called positive and negative cases. We also give explicit examples of manifolds with warped product ends and…
We prove that the Calabi-Yau equation can be solved on the Kodaira-Thurston manifold for all given $T^2$-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic…
We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully…
The theory of coverings of the two-dimensional torus is a standard part of algebraic topology and has applications in several topics in string theory, for example, in topological strings. This paper initiates applications of this theory to…
We establish the existence of strong solutions to a class of nonlinear strongly coupled and uniform elliptic systems consisting of more than two equations. The existence of of nontrivial and non constant solutions (or pattern formations)…
In this paper, we consider the nonlinear elliptic equations on rectangular tori. Using methods in the study of KAM theory and Anderson localization, we prove that these equations admit many analytic solutions.
We study the initial value problem of fully nonlinear third-order equations on the torus. Under some conditions on the nonlinearity and the data, we prove that the equation behaves like a parabolic one: there exists a unique local solution…
In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case where a sequence of Riemannian manifolds is conformal to a…
In this paper we discuss a class of normal forms of the completely resonant non--linear Schr\"odinger equation on a torus. We stress the geometric and combinatorial constructions arising from this study. Further analytic considerations and…
We study a nonlinear and nonlocal elliptic equation posed on the flat torus. While constant solutions always exist, we show that uniqueness fails in general. Using spectral analysis and the Crandall--Rabinowitz bifurcation theorem, we prove…
We prove a priori estimates for a class of transverse fully nonlinear equations on Sasakian manifolds and give some geometric applications such as the transversion Calabi-Yau theorem for transverse balanced and (strongly) Gauduchon metrics.…
We generalize the Tian-Todorov Theorem in the case of Calabi-Yau varieties equipped with a line bundle.