arXiv research

Alexandrov immersed surfaces maintain their properties under mean curvature flow.

We show that any minimal torus in $S^3$ which is Alexandrov immersed must be rotationally symmetric. An analogous result holds for surfaces of constant mean curvature.

Two families of genus one surfaces with $H$-curvature in $\mathbb{H}^2 imes\mathbb{R}$ are constructed.

We prove that compact 3-manifolds $M$ of constant curvature +1 with boundary a minimal surface are locally naturally parametrized by the conformal class of the boundary metric $γ$ in the Teichmuller space of $\partial M$, when $genus(\partial M) \geq 2$. Stronger results are obtained in the case of genus 1 boundary, gi…

In this survey, we discuss various aspects of the minimal surface equation in the three-sphere S^3. After recalling the basic definitions, we describe a family of immersed minimal tori with rotational symmetry. We then review the known examples of embedded minimal surfaces in S^3. Besides the equator and the Clifford t…