Related papers: Uniqueness type result in dimension 3

The Manin conjecture is established for a split singular cubic surface in P^3, with singularity type D_5.

The paper studies the problem of prescribing positive cross curvature on the three-dimensional sphere. We produce several existence results and an example of non-uniqueness, disproving a conjecture of Hamilton's.

We give construction of singular K3 surfaces with discriminant 3 and 4 as double coverings over the projective plane. Focusing on the similarities in their branching loci, we can generalize this construction, and obtain a three dimensional…

A warped product with a spherical factor and a logarithmically concave warping function satisfies a scalar curvature rigidity of the Llarull type. We develop a scalar curvature rigidity of the Llarull type for a general class of domains in…

We revisit some uniqueness results for a geometric nonlinear PDE related to the scalar curvature in Riemannian geometry and CR geometry. In the Riemannian case we give a new proof of the uniqueness result assuming only a positive lower…

We give an explicit formula for singular surfaces of revolution with prescribed unbounded mean curvature. Using it, we give conditions for singularities of that surfaces. Periodicity of that surface is also discussed.

We prove subelliptic estimates for the dbar-problem at the isolated singularity of the variety $z^2=xy$ in $\mathbb{C}^3$.

We study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension $4$, and an existence theorem which holds in dimensions $n \geq 4$. This problem is…

All varieties, extremal contractions, singularities are divided on exceptional and non-exceptional ones. Roughly speaking, there are the infinite families of non-exceptional varieties, extremal contractions or singularities and only the…

The unique third-order invariant variational equation in three-dimensional (pseudo)Euclidean space is derived.

We obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.

As indicated by the third author in [19], there is a gap in the previous version of this paper by the first two authors [5]. We provide in this version an argument to fix the aforementioned gap. The main proposition, whose proof uses…

We develop a method for proving sup-norm and H\"older estimates for $\overline{\partial}$ on wide class of finite type pseudoconvex domains in $\mathbb{C}^n$. A fundamental obstruction to proving sup-norm estimates is the possibility of…

We resolve a conjecture of F\"assler and Orponen on the dimension of exceptional projections to one-dimensional subspaces indexed by a space curve in $\mathbb{R}^3$. We do this by obtaining sharp $L^p$ bounds for a variant of the Wolff…

We give blow-up analysis for the solutions of an elliptic equation under some conditions. Also, we derive a compactness result for this equation.

We give a sup $\times$ inf inequality for an elliptic equation.

We obtain estimates for maximal functions that arise when one studies Nikodym-type sets. We also formulate a curvature condition that allows favorable estimates for these maximal functions.

We give necessary and sufficient conditions on the singular Bj\"{o}rling data to the singular Bj\"{o}rling problem's solution has a prescribed nature of singularity. As an application, we prove that near a maxface with a particular type of…

For singular corank 1 surfaces in $\mathbb R^3$ we introduce a distinguished normal vector called the axial vector. Using this vector and the curvature parabola we define a new type of curvature called the axial curvature, which generalizes…

In this paper we develop methods to extend the minimal hypersurface approach to positive scalar curvature problems to all dimensions. This includes a proof of the positive mass theorem in all dimensions without a spin assumption. It also…