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Register a new userA rigidity theorem for ideal surfaces with flat boundary
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We consider surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation corresponding to extremising the $L^2$-norm of the gradient of the mean curvature. We show that such surfaces with small $L^2$-norm of the second fundamental form and satisfying so-called `flat boundary conditions are necessarily planar.
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In this note, we study minimal Lagrangian surfaces in $mathbb{B}^4$ with Legendrian capillary boundary on $mathbb{S}^3$. On the one hand, we prove that any minimal Lagrangian surface in $mathbb{B}^4$ with Legendrian free boundary on $mathbb{S}^3$ must be an equatorial plane disk. One the other hand, we show that any annulus type minimal Lagrangian surface in $mathbb{B}^4$ with Legendrian capillary boundary on $mathbb{S}^3$ must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng (Sci. China Math., 2020).
Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the ONeill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a Kahler-Einstein manifold or a Riemannian product of a Kahler-Einstein manifold with R (or with the circle S^1).
An analogue of the Stefan-Sussmann Theorem on manifolds with boundary is proven for normal distributions. These distributions contain vectors transverse to the boundary along its entirety. Plain integral manifolds are not enough to integrate a normal distribution; the next best integrals are so-called neat integral manifolds with boundary. The conditions on the distribution for this integrability is expressed in terms of adapted collars and integrability of a pulled-back distribution on the interior and on the boundary.
The aim of this work is to present an initial data version of Hawkings theorem on the topology of back hole spacetimes in the context of manifolds with boundary. More precisely, we generalize the results of G. J. Galloway and R. Schoen [13] and G. J. Galloway [11, 12] by proving that a compact free boundary stable marginally outer trapped surface (MOTS) $Sigma$ in an initial data set with boundary satisfying natural dominant energy conditions (DEC) is of positive Yamabe type, i.e. $Sigma$ admits a metric of positive scalar curvature with minimal boundary, provided $Sigma$ is outermost. To do so, we prove that if $Sigma$ is a compact free boundary stable MOTS which does not admit a metric of positive scalar curvature with minimal boundary in an initial data set satisfying the interior and the boundary DEC, then an outer neighborhood of $Sigma$ can be foliated by free boundary MOTS $Sigma_t$, assuming that $Sigma$ is weakly outermost. Moreover, each $Sigma_t$ has vanishing outward null second fundamental form, is Ricci flat with totally geodesic boundary, and the dominant energy conditions saturate on $Sigma_t$.
We employ min-max techniques to show that the unit ball in $mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
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