The minimal area covered by the image of the unit disk is found for nonvanishing univalent functions normalized by the conditions f(0) = 1, f ′(0) = α. Two different approaches are discussed, each of which contributes to the complete solution of the problem. The first approach reduces the problem, via symmetrization, to the class of typically real functions, where the well-known integral repres...

We show that the Holmes--Thompson area of every Finsler disk radius $r$ whose interior geodesics are length-minimizing is at least $\frac{6}{\pi} r^2$. Furthermore, we construct examples showing inequality sharp and observe equality case attained by a non-rotationally symmetric metric. This contrasts with Berger's conjecture in Riemannian case, which asserts round hemisphere extremal. To prove ...

This paper mainly concerns the area growth and bottom spectrum of complete stable minimal surfaces in a three-dimensional manifold with scalar curvature bounded from below. When ambient is Euclidean space, by an elementary argument, it shown directly stability inequality that such grows exactly as plane. Consequently, must be flat, well-known result due to Fisher-Colbrie Schoen well do Carmo Pe...

Let $\Omega\subset\Bbb{R}^{n+1}$ have minimal Gaussian surface area among all sets satisfying $\Omega=-\Omega$ with fixed volume. $A=A_x$ be the second fundamental form of $\partial\Omega$ at $x$, i.e., $A$ is matrix first order partial derivatives unit normal vector $x\in\partial\Omega$. For any $x=(x_1,\ldots,x_{n+1})\in\Bbb{R}^{n+1}$, let $\gamma_n(x)=(2\pi)^{-n/2}e^{-(x_1^2+\cdots+x_{n+1}^2...