Parabolicity, projective volume and finiteness of total curvature of minimal surfaces (Q1772503)

The author considers a minimal surface \({\mathbf x}\:M\to {\mathbb R}^3\). It is assumed that \(M\) is stochastically complete which means that \[ \int_M p(t,x,y)\,dv(y) =1, \quad\text{for }\;x\in M \,. \] Here \(p(t,x,y)\) is the heat kernel for \((1/2)\Delta_M\) and \(dv\) is the measure defined by Riemannian volume. The projective volume of the minimal surface is measured by \[ Q_p(M,\mathbf{a}) = \int_M \frac{| (\text\textbf{x}-\text\textbf{a})^\perp | ^2} {| \text\textbf{x}-\text\textbf{a}| ^{4}}\,dv \,, \] where \((\mathbf{x}-\text\textbf{a})^\perp(x)\) is a projection of \( \mathbf{x}-\text\textbf{a} \) on the normal space to the surface \( \mathbf{x}(M)\) at a point \(\mathbf{x}(x)\). This function is used because it is well-defined even if the immersion is not proper. The main result states that if the minimal surface has a finite number of ends and finite genus and if \(Q_p(M,\mathbf{a})<\infty\) holds for some \(\mathbf{a}\) at positive distance from \(\mathbf{x}(M)\), then \(M\) is of finite total curvature. The main result is a generalization of a result of \textit{Q. Chen} [Manuscr. Math. 92, No. 2, 135--142 (1997; Zbl 0884.53010)] which applies to properly immersed minimal surfaces with quadratic volume growth. An important tool for the proof of the main result is a lemma concerning the logarithmic derivative of subharmonic functions.