Related papers: The dual Minkowski problem for negative indices
In \cite{LX}, the first author and the third author introduced and studied the horospherical $p$-Minkowski problem for smooth horospherically convex domains in hyperbolic space. In this paper, we introduce and solve the discrete…
Pairs of metrics in a three-dimensional linear vector space are considered, one of which is a Minkowski type metric with the signature (+,-,-). Such metric pairs are classified and canonical presentations for them in each class are…
In [Calc. Var., 57:5 (2018)], Hong-Ye-Zhang proposed the $p$-capacitary Orlicz-Minkowski problem and proved the existence of convex solutions to this problem by variational method for $p\in(1,n)$. However, the smoothness and uniqueness of…
In this paper, we first study invariants of curves parametrized by a real variable in the dual plane $\mathbb{D}^2$ under equiaffine transformations. We then obtain explicit equations for all curves in $\mathbb{D}^2$ whose equiaffine…
In the year 1990, B\'ela Bollob\'as, Imre Leader and Andrew Radcliffe considered the following combinatorial problem: given three parameters k, n and q, find a set of k vertices in the binary n-cube which contains a maximal number of…
Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \| f(z) \|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty…
We provide a complete study of existence and uniqueness of solutions to the Lichnerowicz equation in general relativity with arbitrary mean curvature.
We study the nonlinear stability of the $(3+1)$-dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the…
We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature…
We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of the $n$-dimensional Euclidean space. For the existence part we…
Peres and Solomyak proved that on $\mathbb R^n$, the limits defining the $L^q$-dimension for any $q\in(0,\infty)\setminus\{1\}$, and the entropy dimension of a self-conformal measure exist, without assuming any separation condition. By…
We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (fourth-order) $Q$-curvature on compact and noncompact manifolds of dimension $\geq5$. Infinitely many branches of metrics with…
We propose a general approach to classification problems in algebraic geometry via mirror duality. For Fano threefolds, a modularity conjecture describes small quantum cohomology and predicts the values of certain Gromov-Witten invariants.
We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is…
In this paper we study the problem of prescribing fractional $Q$-curvature of order $2\sigma$ for a conformal metric on the standard sphere $\Sn$ with $\sigma\in (0,n/2)$ and $n\geq2$. Compactness and existence results are obtained in terms…
Rubin's generalized Minkowski--Funk transforms $M_t^\alpha$ on the sphere $\mathbb{S}^n$ give rise, for irrational radii $t=\cos(\beta\pi)$, to a small denominator problem governed by the asymptotic behavior of their spectral multipliers.…
We prove a collection of reverse Alexandrov-Fenchel type inequalities in anisotropic, Euclidean, spherical, and hyperbolic settings. The unifying principle is that the relevant deficit is controlled by curvature radius data, or equivalently…
Gravity is not only able to be mimicked in flat spacetimes, but also in curved spacetimes. We study analogue gravity models in curved spacetime by considering the relativistic Gross-Pitaevskii theory and Yang-Mills theory in the fixed…
Hermann Minkowski introduced a function in 1904 which maps quadratic irrational numbers to rational numbers and this function is now known as Minkowski's question mark function since Minkowski used the notation $?(x)$. This function is a…
Necessary and sufficient conditions for the existence of solutions to the asymmetric $L_p$ Minkowski problem in $\mathbb{R}^2$ are established for $0 < p < 1$.