Related papers: Li-Yau type inequality for curves in any codimensi…

We show that any asymptotically locally Euclidean (ALE) metric which is obstruction-flat or extended obstruction-flat must be ALE of a certain optimal order. Moreover, our proof applies to very general elliptic systems and in any dimension…

In this paper, we study the problem of energy conservation for the solutions to the incompressible viscoelastic flows. First, we consider Leray-Hopf weak solutions in the bounded Lipschitz domain $\Omega$ in $\mathbb{R}^d\,\, (d\geq 2)$. We…

We classify all finite energy solutions of an equation which arises as the Euler--Lagrange equation of a conformally invariant logarithmic Sobolev inequality on the sphere due to Beckner. Our proof uses an extension of the method of moving…

We derive a parabolic version of Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on $\ell$-sectional curvature. We apply this to show that the image of Gauss map is preserved under a…

This paper is concerned with qualitative properties of bounded steady flows of an ideal incompressible fluid with no stagnation point in the two-dimensional plane R^2. We show that any such flow is a shear flow, that is, it is parallel to…

We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we…

Let $(M, g)$ be an dimensional complete Riemannian manifold. In this paper we prove local Li-Yau type gradient estimates for all positive solutions to the following nonlinear parabolic equation \begin{equation*} (\partial_t - \Delta_g +…

In this work we prove a bound for the torsion in Mordell-Weil groups of smooth elliptically fibered Calabi-Yau 3- and 4-folds. In particular, we show that the set which can occur on a smooth elliptic Calabi-Yau $n$-fold for ($n\geq 3$) is…

We provide the first explicit example of Type IIB string theory compactification on a globally defined Calabi-Yau threefold with torsion which results in a four-dimensional effective theory with a non-Abelian discrete gauge symmetry. Our…

Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface $H$ so that restrictions $\phi_j |_H$ to $H$ of $\Delta$-eigenfunctions of Riemannian manifolds $(M, g)$ with ergodic geodesic flow are quantum ergodic…

We investigate the lower bound for higher eigenvalues $\lambda_i$ of the poly-Laplace operator on a bounded domain and improve the famous Li-Yau inequality and its related results. Firstly, we consider the low dimensional cases for the…

We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We…

Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.

The algebraic properties of drift-flux two-phase fluids models without gravitational and wall friction forces are studied. More precisely, for the two fluids we consider equation of states of polytropic gases. We perform a classification…

We obtain the Phragm\`en-Lindel\"of principle on combinatorial infinite weighted graphs for the Cauchy problem associated to a certain class of parabolic equations with a variable density. We show that the hypothesis made on the density is…

We reprove Wolff's $L^{\frac{5}2}-$ bound for the $\R^3-$Kakeya maximal function without appealing to the argument of induction on scales. The main ingredient in our proof is an adaptation of Sogge's strategy used in the work on…

Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their…

We introduce a curvature-dimension condition for autonomous Lagrangians on weighted manifolds, which depends on the Euler-Lagrange dynamics on a single energy level. By generalizing Klartag's needle decomposition technique to the Lagrangian…

We study the quantum geometry of the class of Calabi-Yau threefolds, which are elliptic fibrations over a two-dimensional toric base. A holomorphic anomaly equation for the topological string free energy is proposed, which is iterative in…

We establish the optimal convergence rate to the hypersonic similarity law, which is also called the Mach number independence principle, for steady compressible full Euler flows over two-dimensional slender Lipschitz wedges. The problem can…