Curvature estimates for \(\mu\)-stable \(G\)-minimal surfaces (Q2763617)

The main result of this dissertation is an a priori estimate of the principal curvatures of a certain class of \(\mu\)-stable \(G\)-minimal surfaces in \(\mathbb R^3,\) which is then used to prove a Bernstein-type theorem for that class of surfaces.NEWLINENEWLINEThe work is divided into four chapters. The necessary differential-geometric background is set in the first chapter. The objects of study are regular surfaces \(\mathfrak x (u, v)\) in \(\mathbb R^3\) defined over a unit open disc \(B\) in \(\mathbb R^2\) which are continuous on the closure \(\bar B\) and of sufficiently high regularity on \(B.\) A new metric is defined by warping an induced metric with a symmetric \(3\times 3\) matrix \(G,\) called the matrix of weights, in the following way: Let \(\mathfrak n\) denote the unit normal vector to a surface and let \(G = G(\mathfrak y) = (g_{ij}(y^k))_{i, j = 1, 2, 3}\) be a symmetric, positive definite matrix which for every nonzero vector \(\mathfrak y \in \mathbb R^3\) satisfies (1) \, \(G(\lambda \mathfrak y) = G(\mathfrak y),\) for all \(\lambda \in (0, \infty);\) (2) \, \(G(\mathfrak y)\circ \mathfrak y ^t = \mathfrak y^t ; \) (3) \, \(G\) is positive definite in accordance with the condition \((1 + g_0)^{-1}| \mathfrak z| ^2 \leq \mathfrak z \circ G(\mathfrak y) \circ \mathfrak z^t \leq (1 + g_0) | \mathfrak z| ^2, \,\) for every \(\mathfrak z \in \mathbb R^3 ,\) with a weight-constant \(g_0 \geq 0;\) (4) \, \(\text{det} \, G(\mathfrak y) = 1.\) Then the new (weighted) metric \(ds^2_g = \mathbf H (u, v)\) is defined by means of the coefficients of the first weighted fundamental form NEWLINE\[NEWLINE h_{ij}(u, v):= \mathfrak x_{u^i}(u, v) \circ G(\mathfrak n) \circ \mathfrak x_{u^j}(u, v)^t, NEWLINE\]NEWLINE where \((u^1, u^2):= (u, v) \in B.\) For two continuously differentiable functions \(\phi\) and \(\psi\) defined on the surface \(\mathfrak x = \mathfrak x (u^1, u^2)\) one defines NEWLINE\[NEWLINE \bar\nabla _{ds^2_g}(\phi , \psi):= h^{ij}(u^1, u^2) \phi _{u^i}(u^1, u^2)\psi_{u^j}(u^1, u^2). NEWLINE\]NEWLINE Then a regular surface \(\mathfrak x : \bar B \to \mathbb R^3\) is called a \(G\)-minimal surface if, when equipped with a weighted metric \(ds^2_g\) it satisfies the parameter-invariant equation \(\bar\nabla _{ds^2_g}(\mathfrak x , \mathfrak n) = 0 \,\) on \(B.\) The author proves several analytical and geometric properties of \(G\)-minimal surfaces. For example, if \(I_G(\mathfrak x)\) and \(III_G(\mathfrak x)\) denote the first and the third weighted fundamental form of the surface, respectively, then the following generalization of a well-known result for minimal surfaces holds: \(III_G(\mathfrak x) + K(u, v)\, I_G(\mathfrak x) = 0, \, \) where \(K\) denotes the Gauss curvature.NEWLINENEWLINEIn the second chapter the author introduces the notion of \(\mu\)-stability. A \(G\)-minimal surface is called \(\mu\)-stable \((\mu > 0)\) if for every function \(\psi \in C_0^\infty (B, \mathbb R)\) the following inequality holds: NEWLINE\[NEWLINE \iint _B \{ \bar\nabla _{ds^2_g}(\psi , \psi) + \mu K \psi ^2 \} \, W \, du\, dv \geq 0, NEWLINE\]NEWLINE where \(W\) denotes the surface-area element. The author outlines two ways to produce \(\mu\)-stable surfaces. One is to start with a stable surface equipped with a weighted metric which turns it into a \(G\)-minimal surface and figure out under what conditions the second variation of the functional \(\iint _B \, J\, (\mathfrak x_u \wedge \mathfrak x_v) \, du dv, \;\) (\(J\) being a suitable density) can be transformed into a \(\mu\)-stability condition. Another way is to derive the \(\mu \)-stability condition from the knowledge of the ``size of the spherical image'' along the lines of the paper of \textit{J. L. Barbosa} and \textit{M. P. do Carmo} [Am. J. Math. 98, 515--528 (1976; Zbl 0332.53006)].NEWLINENEWLINEThe third chapter deals with several interesting consequences of \(\mu \)-stability such as an estimate of the energy of the position vector: NEWLINE\[NEWLINE \frac{1}{r^2} \iint _B \{ | \mathfrak x _u (u, v)| ^2 + | \mathfrak x _v (u, v)| ^2\}\, du \,dv \leq \; \text{const}\, (\mu), NEWLINE\]NEWLINE where \(r\) represents the geodesic radius of the surface.NEWLINENEWLINEThe main results are presented in the fourth chapter, where the following estimate for \(\mu \)-stable \(G\)-minimal surfaces with \(\mu > (1+g_0)/2, \,\) represented as a geodesic disc of radius \(r\) around the center \(\mathfrak x (0, 0), \,\) is proved: NEWLINE\[NEWLINE k_1 (0, 0)^2 + k_2 (0, 0)^2 \leq \frac{1}{r^2}\, \Theta, NEWLINE\]NEWLINE where \(k_1, k_2\) are the principal curvatures and \(\Theta\) is an a priori constant depending on \(\mu\) and the class of the surface considered. This result is then used to prove a Bernstein-type theorem: A complete \(\mu \)-stable \(G\)-minimal surface \(\mathfrak x\) with \(\mu > (1+g_0)/2 ,\) defined over the entire plane \(\mathbb R^2\) represents a linear function.