3 citations · 5 across the 9 of their papers we have counts for
9 papers
Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces
Zhenhua Liu
We prove that area-minimizing submanifolds in mod homology are not generically smooth, except in the case of geodesics, minimal surfaces and minimal hypersurfaces. This answers…
Area-minimizing surfaces with fractal singular sets: prevalence, moduli space, and refinements on strata
Zhenhua Liu
This paper is a continuation of our work on a conjecture of Almgren on area-minimizing surfaces with fractal singular sets. First, we prove that area-minimizing surfaces with fract…
Area-minimizing submanifolds are not generically smooth
Zhenhua Liu
We prove that area-minimizing submanifolds are not generically smooth, settling a conjecture of White that asks the generic smoothness of area-minimizing submanifolds. We furthermo…
Every finite graph arises as the singular set of a compact -d calibrated area minimizing surface
Zhenhua Liu
Given any (not necessarily connected) combinatorial finite graph and any compact smooth -manifold with the third Betti number , we construct a calibrated 3-dime…
Weak Counterexamples to Curvature Estimates for Minimizing Surfaces
Zhenhua Liu
We construct a sequence of smooth minimizing surfaces in a sequence of metrics converging to the standard Euclidean metric, so that they have diverging norm of second fundame…
Stationary one-sided area-minimizing hypersurfaces with isolated singularities
Zhenhua Liu
We extend the results of Hardt and Simon on area-minimizing cones to prove that isolated singularities of stationary one-sided area-minimizing hypersurfaces can be locally perturbe…
Proof of Bishop's volume comparison theorem using singular soap bubbles
Hubert Bray, Feng Gui, Zhenhua Liu +1
Bishop's volume comparison theorem states that a compact -manifold with Ricci curvature larger than the standard -sphere has less volume. While the traditional proof uses geo…
The Existence of Embedded -Invariant Minimal Hypersurface
Zhenhua Liu
For a compact connected Lie group acting as isometries on a compact orientable Riemannian manifold and cohomogeneity not equal to 0 or 2, we prove the existence of a…
Stability Inequalities for Lawson Cones
Zhenhua Liu
We prove stability inequalities for Lawson cones with This extends the results of G. De. Philippis and F. Magg…