On $\delta$-Stable Minimal Hypersurfaces in $\mathbb{R}^{n+1}$

Thu, 04 Jul 2024 11:51:14 +0800

Main results

The paper extends regularity, compactness, and Bernstein theory from stable minimal hypersurfaces to the weaker notion of $\delta$-stability.

Regularity and compactness

A quantitative theory above the catenoid threshold

For $n\geq3$ and $\delta>(n-2)/n$, sequences with uniform area control admit varifold subsequences whose singular sets satisfy an explicit dimension bound determined by $\delta$.

Bernstein theorem

Flatness in dimensions three and four

For $n=3,4$, complete two-sided $\delta$-stable immersed hypersurfaces are flat under either a stronger numerical stability range or the combination of simple connectivity, properness, and finitely many ends.

Sharp threshold

The lower bound on $\delta$ cannot be improved

The $n$-dimensional catenoid is exactly $(n-2)/n$-stable. Simons-type cones also identify the sharp threshold for the cone-classification component of the theory.

Proof strategy

The regularity argument combines a Caccioppoli inequality, De Giorgi iteration, an $\varepsilon$-regularity theorem for the second fundamental form, and tangent-cone analysis. A sharp classification of $\delta$-stable minimal cones supplies the dimension reduction. In low dimensions, point-picking and curvature estimates reduce the Bernstein statements to the global flatness results.

Generalized Bernstein Theorem for Stable Minimal Plateau Surfaces

Thu, 20 Oct 2022 11:51:14 +0800

Main results

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

Stability and quadratic area growth force flatness

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

The surface is cylindrical over a planar network

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

Only two global configurations remain

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.

Proof strategy

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.

Generalized Bernstein Theorem | Gaoming Wang

On $\delta$-Stable Minimal Hypersurfaces in $\mathbb{R}^{n+1}$

Main results

A quantitative theory above the catenoid threshold

Flatness in dimensions three and four

The lower bound on $\delta$ cannot be improved

Proof strategy

Generalized Bernstein Theorem for Stable Minimal Plateau Surfaces

Main results

Stability and quadratic area growth force flatness

The surface is cylindrical over a planar network

Only two global configurations remain

Proof strategy