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SINGLY-PERIODIC MINIMAL SURFACES IN H2 × R
표준철 대한수학회 2012 대한수학회보 Vol.49 No.5
We construct three kinds of complete embedded singly-periodic minimal surfaces in $\mathbb{H}^2{\times}\mathbb{R}$. The first one is a 1-parameter family of minimal surfaces which is asymptotic to a horizontal plane and a vertical plane; the second one is a 2-parameter family of minimal surfaces which has a fundamental piece of finite total curvature and is asymptotic to a finite number of vertical planes; the last one is a 2-parameter family of minimal surfaces which fill $\mathbb{H}^2{\times}\mathbb{R}$ by finite Scherk's towers.
표준철 영남수학회 2022 East Asian mathematical journal Vol.38 No.3
In this paper, we prove that the first nonzero eigenvalues λ1 of the Laplacian and the p-Laplacian are decreasing along the inverse mean curvature flow with forced term in Euclidean space.
REGULARITY OF SOAP FILM-LIKE SURFACES SPANNING GRAPHS IN A RIEMANNIAN MANIFOLD
Robert Gulliver,박성호,표준철,서검교 대한수학회 2010 대한수학회지 Vol.47 No.5
Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant −κ2. Using the cone total curvature TC(Γ) of a graph ¡ which was introduced by Gulliver and Yamada [8], we prove that the density at any point of a soap film-like surface Σ spanning a graph ¡ Γ M is less than or equal to [수식]. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when n = 3,this density estimate implies that if [수식] then the only possible singularities of a piecewise smooth (M, 0, δ)-minimizing set Σ are the Y -singularity cone. In a manifold with sectional curvature bounded above by b2 and diameter bounded by π/b, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.