TY - JOUR

T1 - One-sided curvature estimates for $H$-disks.

AU - H. Meeks III, William

AU - Tinaglia, Giuseppe

PY - 2020/10/2

Y1 - 2020/10/2

N2 - In this paper we prove a one-sided curvature estimate for disks embedded
in R3 with constant mean curvature. An important feature of this estimate
is its independence on the value of the constant mean curvature.
For clarity of exposition, we will call an oriented surface Σ immersed in R
3 an H-surface if it is embedded, connected and it has non-negative constant
mean curvature H. We will call an H-surface an H-disk if the H-surface
is homeomorphic to a closed unit disk in the Euclidean plane. We remark
that this definition of H-surface differs from the one given in [? ], where we
restricted to the case where H > 0. In this paper B(R) denotes the open
ball in R3 centered at the origin ~0 of radius R, B(R) denotes its closure and
for a point p on a surface Σ in R3, |AΣ|(p) denotes the norm of the second
fundamental form of Σ at p.

AB - In this paper we prove a one-sided curvature estimate for disks embedded
in R3 with constant mean curvature. An important feature of this estimate
is its independence on the value of the constant mean curvature.
For clarity of exposition, we will call an oriented surface Σ immersed in R
3 an H-surface if it is embedded, connected and it has non-negative constant
mean curvature H. We will call an H-surface an H-disk if the H-surface
is homeomorphic to a closed unit disk in the Euclidean plane. We remark
that this definition of H-surface differs from the one given in [? ], where we
restricted to the case where H > 0. In this paper B(R) denotes the open
ball in R3 centered at the origin ~0 of radius R, B(R) denotes its closure and
for a point p on a surface Σ in R3, |AΣ|(p) denotes the norm of the second
fundamental form of Σ at p.

U2 - 10.4310/CJM.2020.v8.n3.a2

DO - 10.4310/CJM.2020.v8.n3.a2

M3 - Article

VL - 8

SP - 479

EP - 503

JO - Cambridge Journal of Mathematics

JF - Cambridge Journal of Mathematics

ER -