arXiv research

We compare various notions of weak subsolutions to degenerate complex Monge-Amp{è}re flows, showing that they all coincide. This allows us to show that the viscosity solution coincides with the envelope of pluripotential subsolutions. Dedicated to Duong Hong Phong on the occasion of his 65th birthday.

The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.

We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…

Study solves complex equation on specific types of manifolds.

We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the …

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…

N. V. Efimov \cite{Ef1} proved that there is no complete, smooth surface in $\R^3$ with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if $M^3$ has sectional curvature between two constants $K_2$ and $K_3$, then there exists $K_1 < \min(K_2, 0)$ such that $M…

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient…

Study of real and quaternionic Lie algebroid connections on manifolds.

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{ä}hler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-D…

We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the g…

Quaternion self-attention reduces computational cost and improves performance.

The paper studies mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.

Quaternionic approach to conformal superminimal surfaces in four-space

Solves complex Monge-Ampère equations on Kähler manifolds.

3D capsule network for point clouds handles rotations and translations.

In this paper, we investigate two hyperbolic flows obtained by adding forcing terms in direction of the position vector to the hyperbolic mean curvature flows in \cite{klw,hdl}. For the first hyperbolic flow, as in \cite{klw}, by using support function, we reduce it to a hyperbolic Monge-Amp$\grave{\rm{e}}$re equation …

Study on quaternionic bisectional curvature for quaternion-Kähler manifolds.

In this paper, we give the definitions and characterizations of quaternionic Salkowski, quaternionic anti-Salkowski and quaternionic similar curves in the Euclidean spaces E^3 and E^4. We obtain relationships between these curves and some special quaternionic curves such as quaternionic slant helices and quaternionic B…

We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. W…

The study finds conditions for quaternionic structures on symmetric spaces.

We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic p…

We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: 1) A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. 2) A map between quaternion…

Defines quaternionic k-vector fields on quaternionic Kähler manifolds.

The paper proves quaternion projective space is unstable.

The paper studies quaternionic structures on GKM graphs and their relation to torus actions on quaternionic projective spaces.

Penrose's two-spinor notation for $4$-dimensional Lorentzian manifolds can be extended to two-component notation for quaternionic manifolds, which is a very useful tool for calculation. We construct a family of quaternionic complexes over unimodular quaternionic manifolds by elementary calculation. On complex quaternio…

We introduce the notion of CR quaternionic map and we prove that any such real-analytic map, between CR quaternionic manifolds, is the restriction of a quaternionic map between quaternionic manifolds. As an application, we prove, for example, that for any submanifold $M$, of dimension $4k-1$, of a quaternionic manifold…

The paper extends Gray's result to quaternion-Kähler manifolds.

Quaternionic curves with specific torsion properties don't exist.

The conformal infinity of a quaternionic-Kahler metric on a 4n-manifold with boundary is a codimension 3-distribution on the boundary called quaternionic contact. In dimensions 4n-1 greater than 7, a quaternionic contact structure is always the conformal infinity of a quaternionic-Kahler metric. On the contrary, in dim…

Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster con…

Quaternionic differential geometry expands geometric concepts using quaternions.

The paper explores quaternionic curves using differential geometry.

Quaternionic Brownian motion on flag manifold linked to sphere diffusion.

Study cohomology of quaternionic foliations and orbifolds.

Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the `quaternionic-like manifolds'. These contain, as particular subclasses, the CR quaternionic and the $ρ$-quaternionic manifolds. Moreover, …

Study counts and equidistributes rational points in quaternionic Heisenberg groups.

We apply the general theory of codimension one integrability conditions for $G$-structures developed in arXiv:1306.6817v3 [math.DG] to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternion…

A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolods. All con…

A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry an…

In this paper we study 3-submersions from a QR-hypersurface of a quaternionic Kaehler manifold onto an almost quaternionic hermitian manifold. We also prove the non-existence of quaternionic submersions between quaternionic Kaehler manifolds which are not locally hyper-Kaehler.

We construct left invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven dimensional…

Study calculates Kulkarni limit sets for quaternionic projective groups.

A quaternionic Kähler manifold M is called {\it positive} if it has positive scalar curvature. The main purpose of this paper is to prove several connectedness theorems for quaternionic immersions in a quaternionic Kähler manifold, e.g. the Barth-Lefschetz type connectedness theorem for quaternionic submanifolds in a p…