An estimate of length for tubular minimal surfaces of arbitrary codimension (Q1204784)

The author studies a minimal immersion \(h: M^ p\to\mathbb{R}^{n+1}\) of a manifold into a Euclidean space which is tubular, i.e. there exist a unit vector \(e\in\mathbb{R}^{n+1}\) and numbers \(-\infty \leq a < b\leq \infty\) such that the set \(M(t_ 1,t_ 2) = \{m\in M: t_ 1 \leq \langle h(m),e\rangle \leq t_ 2\}\) is compact for arbitrary \(t_ 1,t_ 2\in (a,b)\). For such an immersion one can define the length of \(M: L(M) = b - a\), where \((a,b)\) is the maximal interval; and the embrace function of \(M\), \(\rho: (a,b) \to \mathbb{R}^ +\), \[ \rho(t) = \max\{(| h(m)|^ 2 - t^ 2)^{1/2}: m\in M(t_ 1,t_ 2)\}. \] The main result is as follows: Theorem. Let \((M^ p,h)\) be a minimal tubular immersion in \(\mathbb{R}^{n+1}\), \(2 \leq p \leq n\). Then the embrace function \(\rho(t)\) is convex and almost everywhere on \((a,b)\) it satisfies the following differential inequality: \[ p''(t)\rho(t) \geq (p-1)(\rho'{}^ 2(t) + 1). \] In particular, when \(p\geq 3\) one obtains the following estimate: \[ L(M) = b - a\leq 2\rho_ 0\varphi_ p < \infty, \] where \(\rho_ 0 = \inf\{\rho(t): t\in (a,b)\}\), \[ \varphi_ p = \int^ \infty_ 1 (y^{2(p-1)}-1)^{-1/2}dy. \]