Umbilic slopes and cubic Weingarten surfaces
Abstract
We study umbilic slopes and the global classification of cubic Weingarten surfaces. For a smooth surface in Euclidean three-space, a nonconstant principal curvature germ with a unique secant tangent has slope zero, minus one, infinity, an odd integer at least three, or its reciprocal. At a nonconstant umbilic germ, a smooth regular relation between mean and Gaussian curvature supplies such a tangent after a continuous labeling of the principal curvatures; the ordered curvature image has at most two limiting directions, both in the same set. We also classify compact immersed surfaces without boundary of class C4 satisfying κ2=cκ13 in some order at every point, for a fixed real c: each connected component is embedded as an ellipsoid of revolution. The arguments share a classification of homogeneous Hessian identities at an umbilic. Scalar unique continuation treats infinite-order contact in the smooth slope problem, whereas an exact fourth-order growth estimate and angular energy estimates yield the cubic classification.
Related
- Positive Geometries from Cubic Surfaces
- Hierarchical hyperbolicity, ball quotients, and the moduli space of smooth cubic surfaces
- Intrinsic mirror symmetry for orbifold cubic surfaces and the A1 spherical DAHA
- Log Calabi-Yau wall crossing for moduli of cubic surfaces
- The moduli space of cubic surfaces and its orbifold fundamental group
BibTeX
@misc{cheng2026umbilic,
title = {{Umbilic slopes and cubic Weingarten surfaces}},
author = {Haoxuan Cheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19188},
eprint = {2609.19188},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19188},
}
RIS
TY - UNPB
TI - Umbilic slopes and cubic Weingarten surfaces
AU - Haoxuan Cheng
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.19188
UR - https://arxiv.org/abs/2609.19188
AB - We study umbilic slopes and the global classification of cubic Weingarten surfaces. For a smooth surface in Euclidean three-space, a nonconstant principal curvature germ with a unique secant tangent has slope zero, minus one, infinity, an odd integer at least three, or its reciprocal. At a nonconstant umbilic germ, a smooth regular relation between mean and Gaussian curvature supplies such a tangent after a continuous labeling of the principal curvatures; the ordered curvature image has at most two limiting directions, both in the same set. We also classify compact immersed surfaces without boundary of class C^4 satisfying κ_2=cκ_1^3 in some order at every point, for a fixed real c: each connected component is embedded as an ellipsoid of revolution. The arguments share a classification of homogeneous Hessian identities at an umbilic. Scalar unique continuation treats infinite-order contact in the smooth slope problem, whereas an exact fourth-order growth estimate and angular energy estimates yield the cubic classification.
ER -