Curvature estimates for surfaces with bounded mean curvature

TY - JOUR

T1 - Curvature estimates for surfaces with bounded mean curvature

AU - Bourni, Theodora

AU - Tinaglia, Giuseppe

PY - 2012/11

Y1 - 2012/11

N2 - Estimates for the norm of the second fundamental form, $|A|$, play a crucial role in studying the geometry of surfaces. In fact, when $|A|$ is bounded the surface cannot bend too sharply. In this paper we prove that for an embedded geodesic disk with bounded $L^2$ norm of $|A|$, $|A|$ is bounded at interior points, provided that the $W^{1,p}$ norm of its mean curvature is sufficiently small, $p>2$. In doing this we generalize some renowned estimates on $|A|$ for minimal surfaces.

AB - Estimates for the norm of the second fundamental form, $|A|$, play a crucial role in studying the geometry of surfaces. In fact, when $|A|$ is bounded the surface cannot bend too sharply. In this paper we prove that for an embedded geodesic disk with bounded $L^2$ norm of $|A|$, $|A|$ is bounded at interior points, provided that the $W^{1,p}$ norm of its mean curvature is sufficiently small, $p>2$. In doing this we generalize some renowned estimates on $|A|$ for minimal surfaces.

U2 - 10.1090/S0002-9947-2012-05487-0

DO - 10.1090/S0002-9947-2012-05487-0

M3 - Article

SN - 0002-9947

VL - 364

SP - 5813

EP - 5828

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 11

ER -