arXiv research

Study finds non-CMC biconservative hypersurfaces in spheres, proving their existence but not embeddability.

New closed non-CMC biconservative surfaces found in round 3-sphere.

Proves existence of solution to Lichnerowicz equation on non-CMC manifolds.

Study on biconservative surfaces in 4D hyperbolic space, providing extrinsic descriptions.

The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method's construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn…

Biconservative surfaces are surfaces with divergence-free stress-bienergy tensor. Simply connected, complete, non-$CMC$ biconservative surfaces in $3$-dimensional space forms were constructed working in extrinsic and intrinsic ways. Then, one raises the question of the uniqueness of such surfaces. In this paper we give…

We survey some recent results on biconservative surfaces in $3$-dimensional space forms $N^3(c)$ with a special emphasis on the $c=0$ and $c=1$ cases. We study the local and global properties of such surfaces, from extrinsic and intrinsic point of view. We obtain all non-$CMC$ complete biconservative surfaces in $\math…

The conformal method is a technique for finding Cauchy data in general relativity solving the Einstein constraint equations, and its parameters include a conformal class, a conformal momentum (as measured by a densitized lapse), and a mean curvature. Although the conformal method is successful in generating constant me…

We prove nonlinear stability for a large class of solutions to the Einstein equations with a positive cosmological constant and compact spatial topology in arbitrary dimensions, where the spatial metric is Einstein with either positive or negative Einstein constant. The proof uses the CMC Einstein flow and stability fo…

Using the implicit function theorem, we prove existence of solutions of the so-called conformally covariant split system on compact 3-dimensional Riemannian manifolds. They give rise to non-Constant Mean Curvature (non-CMC) vacuum initial data for the Einstein equations. We investigate the conformally covariant split s…

We construct simply connected, complete, non-$CMC$ biconservative surfaces in the $3$-dimensional hyperbolic space $\mathbb{H}^3$ in an intrinsic and extrinsic way. We obtain three families of such surfaces, and, for each surface, the set of points where the gradient of the mean curvature function does not vanish is de…

In the context of the Bartnik mass, there are two fundamentally different notions of an extension of some compact Riemannian manifold $(Ω,γ)$ with boundary. In one case, the extension is taken to be a manifold without boundary in which $(Ω,γ)$ embeds isometrically, and in the other case the extension is taken to be a m…