Ends of immersed minimal and Willmore surfaces in asymptotically flat spaces

A Note on Existence and Non-existence of Minimal Surfaces in Some Asymptotically Flat 3-manifolds

Motivated by problems on apparent horizons in general relativity, we prove the following theorem on minimal surfaces: Let g be a metric on the three-sphere S satisfying Ric(g) ≥ 2g. If the volume of (S, g) is no less than one half of the volume of the standard unit sphere, then there are no closed minimal surfaces in the asymptotically flat manifold (S \ {P}, Gg). Here G is the Green’s function...

Loop Quantum Gravity and Asymptotically Flat Spaces

After motivating why the study of asymptotically flat spaces is important in loop quantum gravity, we review the extension of the standard framework of this theory to the asymptotically flat sector detailed in [1]. In particular, we provide a general procedure for constructing new Hilbert spaces for loop quantum gravity on non-compact spatial manifolds. States in these Hilbert spaces can be int...

Ju n 20 16 A NON - EXISTENCE RESULT FOR MINIMAL CATENOIDS IN ASYMPTOTICALLY FLAT SPACES

In General Relativity, asymptotically flat manifolds naturally arise as spacelike slices in space-times describing isolated gravitational systems. Due to their self-evident physical relevance, their geometry has been extensively studied and a variety of fundamental results have been obtained. Among these, the mean-curvature proof of the Positive Mass Theorem proposed by Schoen and Yau [SY79] ha...

A non-existence result for minimal catenoids in asymptotically flat spaces

We show that asymptotically Schwarzschildean 3-manifolds cannot contain minimal surfaces obtained by perturbative deformations of a Euclidean catenoid, no matter how small the ADM mass of the ambient space and how large the neck of the catenoid itself. Such an obstruction is sharply three-dimensional and ceases to hold for more general classes of asymptotically flat data.

On helicoidal ends of minimal surfaces