Classification of Complete Regular Minimal Surfaces in ℝn with Total Curvature −6π
Abstract
:1. Introduction
- (a)
- S is a regular complex analytic curve in defined on with the maximal degree 4 at the pole ∞ of its components;
- (b)
- S is a regular complex analytic curve in defined on with the maximal degree at the poles respectively of its components;
- (c)
- Up to a constant factor,
- (d)
- Up to a constant factor,
2. Preliminaries
- (a).
- M is conformally to a compact Riemann surface with a finite number, say r, points deleted;
- (b).
- C is an even integer, and satisfies
- (c).
- The Gauss map of S extends to a map of whose image is an algebraic curve in lying in ; the total curvature of S is equal in absolute value to the area of , counting multiplicities;
- (d).
- intersects a fixed finite number m times (counting multiplicity) every hyperplane in except for those hyperplanes containing one or more of the (finite number of) points of .
- (i)
- Each is a polynomial;
- (ii)
- The maximal degree of the is m;
- (iii)
- The have no common factor;
- (iv)
- .
- (i)
- Each is a polynomial;
- (ii)
- The maximum degree of the is m;
- (iii)
- The have no common factor;
- (iv)
- .
- 1.
- S is simply-connected with ;
- 2.
- S is doubly-connected with
- 3.
- and S is a holomorphic curve in
3. Proof of the Main Theorems
4. A Method to Give Some Families of Examples
- Step 1. Fix a series of data with . To simplify the next step, we could just choose and others arbitrarily.
- Step 1. Fix a series of data with , , others arbitrarily.
5. Complete Minimal Surfaces with Total Curvature in
5.1. Further Structure of
5.2. Proof of Theorem 3
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Li, M.; Ou, L.; Zhang, S. Classification of Complete Regular Minimal Surfaces in ℝn with Total Curvature −6π. Mathematics 2022, 10, 1820. https://doi.org/10.3390/math10111820
Li M, Ou L, Zhang S. Classification of Complete Regular Minimal Surfaces in ℝn with Total Curvature −6π. Mathematics. 2022; 10(11):1820. https://doi.org/10.3390/math10111820
Chicago/Turabian StyleLi, Minghao, Li Ou, and Shuangcai Zhang. 2022. "Classification of Complete Regular Minimal Surfaces in ℝn with Total Curvature −6π" Mathematics 10, no. 11: 1820. https://doi.org/10.3390/math10111820
APA StyleLi, M., Ou, L., & Zhang, S. (2022). Classification of Complete Regular Minimal Surfaces in ℝn with Total Curvature −6π. Mathematics, 10(11), 1820. https://doi.org/10.3390/math10111820