In this paper, we consider a Generalized Bernstein Theorem for a type of generalized minimal surfaces, namely minimal Plateau surfaces. We show that if a complete orientable minimal Plateau surface is stable and has quadratic area growth in $R^3$, then it is flat.
The classical Bernstein question is extended from smooth minimal surfaces to Plateau surfaces, whose local models may also contain $Y$- and $T$-type soap-film singularities.
Generalized Bernstein theorem
If a complete orientable minimal Plateau surface in $\mathbb R^3$ is stable and has at most quadratic area growth, then every component of its regular part lies in a plane.
No $T$-points
When no $T$-type singularity occurs, the surface is, after a rigid motion, of the form $N\times\mathbb R$ for an embedded stationary network $N\subset\mathbb R^2$.
With $T$-points
If $T$-type singularities occur, there is either one such point and the surface is the standard $T$ cone, or there are exactly two points joined in the explicitly classified flat configuration.
The proof transfers the stability inequality to the regular sheets while retaining the balancing information along the singular curves. Curvature estimates and logarithmic cutoffs, made possible by quadratic area growth, force the second fundamental form to vanish. Once each sheet is planar, the topology and Plateau angle conditions reduce the remaining work to a global classification of the straight $Y$-curves and isolated $T$-points.
My research interests include Geometric Analysis and Partial Differential Equations.