A Wintgen type inequality for surfaces in 4D neutral pseudo-Riemannian space forms and its applications to minimal immersions

Publication date: 11 July 2013

Abstract: Let

M

be a space-like surface immersed in a 4-dimensional pseudo-Riemannian space form

R_{2}^{4} (c)

with constant sectional curvature

c

and index two. In the first part of this article, we prove that the Gauss curvature

K

, the normal curvature

K^{D}

, and mean curvature vector

H

of

M

satisfy the general inequality:

K + K^{D} g e q H, H + c

. In the second part, we investigate space-like minimal surfaces in

R_{2}^{4} (c)

which satisfy the equality case of the inequality identically. Several classification results in this respect are then obtained.

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