arXiv research

Generalizes Collins' theorem to products of locally indicable groups.

We prove a priori estimates for a generalised Monge-Ampère PDE with "non-constant coefficients" thus improving a result of Sun in the Kähler case. We apply this result to the deformed Hermitian Yang-Mills (dHYM) equation of Jacob-Yau to obtain an existence result and a priori estimates for some ranges of the phase angl…

Classifies Ricci collineations on specific 3D Lorentzian Lie groups.

Study finds all conformal Ricci collineations on specific 3D Lorentzian groups.

Study finds all Ricci collineations for specific connections on 3D Lorentzian groups.

The study classifies special flows on specific geometric groups.

Considering prolongation of a Lie algebroid equipped with a spray, defining some classical tensors, we show that a Lie symmetry of a spray is a curvature collineation for these tensors.

We establish $C^{2,α}$ estimates for PDE of the form convex $+$ a sum of weakly concave functions of the Hessian, thus generalising a recent result of Collins which is in turn inspired by a theorem of Caffarelli and Yuan. Independently, we also prove an existence result for a certain generalised Monge-Ampère PDE.

Period maps surjective for certain gravitational instantons.

Let $F$ be Cayley's ruled cubic surface in a projective three-space over any commutative field $K$. We determine all collineations fixing $F$, as a set, and all cubic forms defining $F$. For both problems the cases $|K|=2,3$ turn out to be exceptional. On the other hand, if $|K|\geq 4$ then the set of simple points of …

Solves non-Archimedean Calabi-Yau equation on complex log pairs.

In this paper we extend a recent result of Collin-Rosenberg ({\it a solution to the minimal surface equation in the Euclidean disc has radial limits almost everywhere}) to a large class of differential operators in Divergence form. Moreover, we construct an example (in the spirit of \cite{CR2}) of a minimal graph in $\…

The study examines Ricci solitons and curvature inheritance on Robinson-Trautman spacetimes.

Locally homogeneous Lorentzian three-manifolds with recurrect curvature are special examples of Walker manifolds, that is, they admit a parallel null vector field. We obtain a full classification of the symmetries of these spaces, with particular regard to symmetries related to their curvature: Ricci and matter colline…

The Ricci Calabi functional is a functional on the space of Kähler metrics of Fano manifolds. Its critical points are called generalized Kähler Einstein metrics. In this article, we show that the Hessian of the Ricci Calabi functional is non-negative at generalized Kähler Einstein metrics. As its application, we give a…

Study disproves conjecture about Hermitian-Yang-Mills solutions.

The paper examines geometric properties of a unique spacetime model.

The paper extends Nakamaye's theorem to non-closed forms on complex manifolds.

The paper confirms the solvability of a complex equation for a 4D manifold.

It was shown by Bonahon-Otal and Hodgson-Rubinstein that any two genus-one Heegaard splittings of the same 3-manifold (typically a lens space) are isotopic. On the other hand, it was shown by Boileau, Collins and Zieschang that certain Seifert manifolds have distinct genus-two Heegaard splittings. In an earlier paper, …

Six quaternionic lines with optimal angles found in 2D quaternion space.

Characterizes Q-Gorenstein singularities via K-stability.

Study resolves conjectures on hypercritical deformed Hermitian-Yang-Mills equation.

New flow solves LYZ equation on Kähler manifolds.

We consider the three-dimensional Heisenberg group, equipped with any left-invariant metric, either Lorentzian or Riemannian. We completely classify their affine vector fields and investigate their relationship with Killing vector fields and their casual character. We also classify their Ricci, curvature and matter col…

Minimal graphs over non-compact domains in 3-manifolds solved with estimates and uniqueness results.

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plan…

The purpose of the present article is to study and characterize sev- eral types of symmetries of generalized Robertson-Walker space-times. Con- formal vector fields, curvature and Ricci collineations are studied. Many im- plications for existence of these symmetries on generalied Robertson-Walker spacetimes are obtaine…

Conditions for Riemannian manifolds to be Euclidean spheres or spaces.

For $0\leq H< 1/2$, we construct entire $H$-graphs in $\mathbb{H}^2\times\mathbb{R}$ that are parabolic and not invariant by one parameter groups of isometries of $\mathbb{H}^2\times\mathbb{R}$. Their asymptotic boundaries are $(\partial_\infty\mathbb{H}^2)\times\mathbb{R}$; they are dense at infinity. When $H=0$ the e…

Characterizes K-semistability for log Fano cone singularities.

This is the second in a series of papers studying the relationship between Rohlin's theorem and gauge theory. We discuss an invariant of a homology S^1 cross S^3 defined by Furuta and Ohta as an analogue of Casson's invariant for homology 3-spheres. Our main result is a calculation of the Furuta-Ohta invariant for the …

Authors discuss complex and non-Archimedean geometry, proving a conjecture.

Consider a finite dimensional (generally reducible) polynomial representation ρof GL_n. A projective compactification of GL_n is the closure of ρ(GL_n) in the space of all operators defined up to a factor (this class of spaces can be characterized as equivariant projective normal compactifications of GL_n). We give an …

Solves critical LYZ equation in Kähler geometry.

Study deformed Hermitian-Yang-Mills equation on complex projective space blowup.

Survey of geometric flows from unified string theories.

Researchers match complex affine structures in mirror constructions.

We present a flexible approach for the valuation of interest rate derivatives based on Affine Processes. We extend the methodology proposed in Keller-Ressel et al. (2009) by changing the choice of the state space. We provide semi-closed-form solutions for the pricing of caps and floors. We then show that it is possible…

Gravitational instantons collapse to a punctured plane with a special Kahler metric.

Introduces a new PDE involving differential forms for Kähler geometry.

Riemannian geometrical tools, such as Ricci collineations and Killing symmetries, so often used in Einstein general theory of gravitation are here applied to plasma physics to build magnetic surfaces from Einstein plasma metrics used in tokamak devices. It is shown that the Killing symmetries are constrains the Einstei…

Paper confirms conjecture for projective manifolds in supercritical phase.

We obtain area growth estimates for constant mean curvature graphs in $\mathbb{E}(κ,τ)$-spaces with $κ\leq 0$, by finding sharp upper bounds for the volume of geodesic balls in $\mathbb{E}(κ,τ)$. We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire graphs in $\mathbb{E}(κ…

Proves solvability of general inverse σ_k equations with constant coefficients.

Theory developed for complex Hessian measures on Hermitian manifolds.

The paper proves a function extension on Kähler manifolds.

Defines volume and Monge-Ampère energy on polarized affine varieties.