A sharp bound for the area of minimal surfaces in the unit ball

By S. Brendle. 2011

54 citations recorded by the source corpus.

Let Σ be a k-dimensional minimal surface in the unit ball Bn which meets the boundary ∂Bn orthogonally. We show that the area of Σ is bounded from below by the volume of the unit ball in $${\mathbb{R}^k}$$.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/f80bf87aca393893388566f5ff8b084271739d99

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In Applied Mathematics

1. Daniel Ketover
2. R. Schoen
3. A. Fraser
4. M. Li
5. S. Brendle
6. Hung Tran
7. G. Kokarev
8. Iosif Polterovich

Citations in the corpus

• Free boundary minimal surfaces of unbounded genus (2016, methodology): “f Proposition 2.1 uses a monotonicity formula for free boundary minimal surfaces that was discovered in various guises independently by Fraser-Schoen (Theorem 5.4 in [FS]), Ros-Vergasta [RV], Brendle [B], and obtained in most general form by Volkmann [V]. To state the consequence of Volkmann’s formu”
• Minimal surfaces and eigenvalue problems (2013, methodology): “p to second order, as we prove in Theorem 3.8 below. Finally, a consequence of the conjecture is that |Σ| ≥ |Bk|. This inequality was shown recently by a direct monotonicity-style argument by Brendle [Br]. Theorem 3.8. Let Σ be a k dimensional minimal submanifold in the ball Bn with ∂Σ ⊂ Sn−1 and wi”
• Free boundary minimal surfaces in the unit ball : recent advances and open questions (2019, unknown): “if and only if 2 is an equatorial plane disk. By a direct monotonicity-type argument, Brendle was abled to generalize Fraser and Schoen’s result to any dimension and codimension. Theorem 7.3 (Brendle [5]). Let k be an immersed free boundary minimal submanifold in Bn Then, j kjjBkj and equality holds”