Related papers: Type II Singularities on complete non-compact Yama…

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} \psi = - \Delta_d \psi$ where $\psi$ is the dual 4-form of a $G_2$-structure $\phi$ and $\Delta_d$ is the Hodge Laplacian on forms. This flow preserves the condition of…

We consider the semilinear wave equation with focusing energy-critical nonlinearity in space dimension 5 with radial data. It is known that a solution $(u, \partial_t u)$ which blows up at $t = 0$ in a neighborhood (in the energy norm) of…

We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-M\"{u}ller-Topping that…

In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere $S^{2n+1}$ moving along the integral curves of the Hopf unit vector field. While such solitons must necessarily be minimal if compact, we produce a…

We define Type I singularities for the mean curvature flow associated to a density $\psi$ ($\psi$MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from…

We study compactness of solutions to the Yamabe problem on Riemannian manifolds which are not locally conformally flat.

We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate…

Let (M,g) be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish…

In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily convex, self-similar solutions of a wide class of fully non-linear parabolic flows in $\mathbb{R}^3$. We show this self-similar solutions are the…

We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time by having its mass concentrating near a…

Given any embedded Lagrangian on a four dimensional compact Calabi-Yau, we find another Lagrangian in the same Hamiltonian isotopy class which develops a finite time singularity under mean curvature flow. This contradicts a weaker version…

In a recent paper, we established optimal Liouville-type theorems for conformally invariant second-order elliptic equations in the Euclidean space. In this work, we prove an optimal Liouville-type theorem for these equations in the…

We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In…

We study a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS) subcritical regime, we present a precise blow-up profile exhibited by the flows. In the HLS critical…

We present a novel method to obtain type IIB flux vacua with flat directions at tree level. We perform appropriate choices of flux quanta that induce relations between the flux superpotential and its derivatives. This method is implemented…

We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n>6 and the trace-free part of the second fundamental form is non-zero everywhere on…

In our previous work [PSSW], we showed that the Ricci flow on S^2 whose initial metric has conical singularities \sum_{j=1}^k \beta_j[p_j] converges to a constant curvature metric with conic singularities (in the stable and semi-stable…

We establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary.…

The formation of singularities on a free surface of a conducting ideal fluid in a strong electric field is considered. It is found that the nonlinear equations of two-dimensional fluid motion can be solved in the small-angle approximation.…

In this paper, we study the formation of finite time singularities in the form of super norm blowup for a spatially inhomogeneous hyperbolic system. The system is related to the variational wave equations as those in [18]. The system posses…