Xin Zhou
Cornell University, NY·Mathematics & Statistics
A soap film stretched across a wire loop settles into the shape with the least possible area, and a soap bubble minimizes its area while enclosing a fixed volume. Mathematicians call the first shape a minimal surface and the second a constant mean curvature (CMC) surface.
- Fields
Mathematics & Statistics
- Topics
minimal surfaces
constant mean curvature
CMC surfaces
- Grants
- 1 known grant, through Jun 2028
What they work on
- Prove the existence of closed minimal surfaces with controlled genus, or number of holes, in three-manifolds, with the goal of resolving Yau's Four Minimal Spheres Conjecture.
- Develop min-max theory, which finds surfaces as balance points of area rather than minima, for minimal hypersurfaces in non-generic settings where standard assumptions fail.
- Establish the existence of constant mean curvature (CMC) hypersurfaces, the mathematical model of soap bubbles, across various geometric settings using geometric variational methods.
More about the research
Proving when such surfaces must exist, and what shapes they can take, is a central problem in geometry that also connects to physics.
Why it matters
This is pure mathematics. It aims to settle open questions about which minimal and constant mean curvature surfaces must exist in curved spaces, including Yau's Four Minimal Spheres Conjecture. The tools developed could deepen understanding in geometric analysis and in related areas of mathematics and physics.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Jonathan Zhu
University of Washington, WA·Mathematics & Statistics·Early-career
Analyze singularities of mean curvature flow near cylindrical and conical models, testing uniqueness of blow-up limits when curvature is present in the interior or at the boundary.
Otis Chodosh
Stanford University, CA·Mathematics & Statistics
Study questions connected to stable minimal hypersurfaces, building on the four-dimensional proof that such surfaces are flat, with the goal of resolving the stable Bernstein problem in higher dimensions.
Christos Mantoulidis
Rice University, TX·Mathematics & Statistics
Investigate the structure of singularities, points where curvature blows up, in area-minimizing minimal surfaces and in mean curvature flow, where a surface evolves to reduce its area.
The school
Cornell University: 37 labs on PI Pages, $1.6B a year on research, 537 research doctorates a year.
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