Web  site  of      Francisco  J. López

López Klein bottle

This surface represents the unique complete minimal Klein bottle in R3 with total curvature -8π. This surface is homeomorphic to a Klein bottle minus one point, and corresponds to the simplest complete minimal immersion with this topology. This means that the absolute value of the total curvature of any other complete minimal finitely punctured Klein bottle in R3 is greater than 8π. For details about its discovery, construction and characterization see the papers:

-Francisco J. López: A complete minimal Klein bottle in R3 . Duke Math. Journal, Vol. 71, No. 1, (1993) 23-30.

-Francisco J. López: On complete nonorientable minimal surfaces with low total curvature. Trans. Amer. Math. Soc., Vol. 348, No. 7, July (1996), 2737-2758.

-Francisco J. López, Francisco Martín: Complete minimal surfaces in R3. Publicacions Matemàtiques, Vol. 43 (1999) 341-449.

I am currently working on variational problems related to the area functional in complete flat 3-manifolds. I am also interested in Complex Analysis and Riemann’s surface theory.

Preprints:

-Proper harmonic embeddings of open Riemann surfaces into R4.  Preprint (with Antonio Alarcón).

Papers:

-Algebraic approximation and the Mittag-Leffler theorem for minimal surfaces, Anal. PDE 15 (2022), no. 3, 859-890. (with Antonio Alarcón).

-Every meromorphic function is the Gauss map of a minimal surface, J. Geom. Anal. 29 (2019), no. 4, 3011-3038. (with A. Alarcon and F. Forstnerič).

-New complex analytic methods in the stufy of non-orientable minimal surfaces in Rn,  Mem. Amer. Math. Soc. 264 (2020), no. 1283, vi+77 pp.

Providence, RI: American Mathematical Society (AMS). (with A. Alarcon and F. Forstnerič).

-Interpolation and optimal hitting for complete minimal surfaces with finite total curvature, Calc. Var. Partial Differential Equations 58 (2019), no. 1, 58:21. (with A. Alarcón e I. Castro-Infantes). Correction, Calc. Var. Partial Differential Equations 59 (2020), no. 5, 59:178. Full tex view only.  ArXiv.

-A construction of complete complex hypersurfaces in the ball with control on the topology, J. Reine Angew. Math. (Crelle's J.) 751 (2019), 289-308. (with A. Alarcón y J. Globevnik).

-Minimal surfaces in minimally convex domains,  Trans. Amer. Math. Soc. 371 (2019), no. 3, 1735-1770 (with A. Alarcón, Barbara Drinovec Drnovšek y F. Forstnerič).

-Holomorphic Legendrian curves, Compos. Math. 153 (2017), no. 9, 1945-1986 (with A. Alarcon y F. Forstnerič).

-Complete bounded embedded complex curves in C2.   J. Eur. Math. Soc., 18 (2016), no. 8, 1675-1705 withn A. Alarcón).

-Embedded minimal surfaces in Rn. Math. Z., 283 (2016) no, 1.2, 1-24 (with A. Alarcon y F. Forstnerič)

-Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves. Proc. London Math. Soc., 111 (2015), no. 4, 851-886 (with A. Alarcón, B. Drinovec Drnovšek y F. Forstnerič).

-Approximation theory for non-orientable minimal surfaces and applications. Geom. and Topol., 19 (2015), no 2, 1015-1062 (with A. Alarcón).

-Complete nonorientable minimal surfaces in R3 and asymptotic behavior. Anal. Geom. Metr. Spaces 2 (2014), 214-234 (with A. Alarcon).

-Properness of associated minimal surfaces. Trans. Am. Math. Soc., 366 (2014), no. 10, 5139-5154 (with A. Alarcón).

-Exotic minimal surfaces. J. Geom. Anal., 24 (2014), no. 2, 988-1006.

-Uniform Approximation by Complete Minimal Surfaces of Finite Total Curvature in R3. Trans. Amer. Math. Soc., 366 (2014), no. 12, 6201-6227.

-Proper holomorphic embeddings of Riemann surfaces with arbitrary topology into C2. J. Geom. Anal., 23 (2013), no. 4, 1794-1805 (with A. Alarcon).

-Compact complete null curves in Complex 3-space. Israel J. Math., 195 (2013), no. 1, 97-122 (with A. Alarcón).

-Harmonic maps and conformal minimal immersions of Riemann surfaces into RN.  Calculus of Variations and PDE, 47 (2013), no. 1-2, 227-242 (with A. Alarcón e I. Fernández).

-On harmonic quasiconformal immersions in R3. Trans. Am. Math. Soc., 365 (2013), no. 4, 1711-1742 (with A. Alarcón).

-Null curves in C3 and Calabi-Yau conjectures. Math. Annalen, 355 (2013), no. 2, 429-455 (with A. Alarcón).

-Minimal surfaces in R3 properly projecting into R2. J. Diff. Geom., 90 (2012), no. 3, 351-382 (with A. Alarcón).

-Nonorientable maximal surfaces in the Lorentz-Minkowski 3-space. Tohoku Math. J. (2) Vol. 62, No. 3 (2010), 311-328 (with Soichi Fujimori).

-Complete minimal surfaces and harmonic functions. Comm. Math. Helv., 87 (2012), n0. 4, 891-904 (with A. Alarcon e I. Fernández).

-On the Uniqueness of the Helicoid and Enneper's surface in the Lorentz-Minkowski space R31 . Trans. Amer. Math. Soc., vol. 363 (2011), 4603-4650 (con I. Fernández).

-Periodic maximal surfaces in the Lorentz-Minkowski space L3. Mathematische Zieitschrift, Vol. 256, 573-601 (2007) (with I. Fernández).

-The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz Minkowski space L3. Manuscripta Mathematica, Vol. 112, 439-463 (2007) (with I. Fernandez y R. Souam).

-Complete nonorientable minimal surfaces in a ball of R3. Trans. Amer. Math. Soc. 358 (2006), 3807-3820 (with S. Morales y F. Martín).

-The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space L3 . Mathematische Annalen, Vol. 332, 605-643 (2005)  (with I. Fernández y R. Souam).

-Some Picard theorems for minimal surfaces. Trans. Amer. Math. Soc., Vol. 356 (2004), no. 2, 703-733.

-Parabolicity and Gauss map of minimal surfaces. Indiana Univ. Math. J., Vol. 52, N. 4, (2003),  1017-1026 (with J. Pérez).

-Adding handles to Nadirashvili's surfaces. J. Diff. Geom., Vol. 60, N. 1, (2002), 155-175 (with F. Martín y S. Morales).

-Minimal surfaces in a Cone. Annals of Global Analysis and Geometry, Vol. 20, n. 3 (2001), 253-299.

-Minimal surfaces in a wegde of a slab. Comm. in Ann. and Geom., Vol. 9, n. 4 (2001), 683-723 (with F. Martín).

-A uniqueness theorem for properly embedded minimal surfaces bounded by straight lines. J. Austral. Math. Soc. (Series A) Vol. 69(2000), 362-402 (with F. Martín).

-Properly immersed minimal disks bounded by straight lines. Math. Ann., Vol. 318 (2000), 667-706 (with F. Wei).

-Maximal surfaces of Riemann type in Lorentz-Minkowski space R3. Michigan J. of Math., Vol. 47 (2000), 469-497 (with R. López y R. Souam).

-A note on the Gauss map of https://www.ugr.es/~fjlopez/_private/gauss.pdfcomplete nonorientable minimal surfaces. Pacific J. of Math., Vol. 194 (2000), No. 1, 129-135.

-Complete minimal surfaces in R3. Publicacions Matemàtiques, Vol. 43 (1999) 341-449 (with F. Martín).

-Properly immersed singly periodic minimal cylinders in R3. Michigan Math. J., Vol. 45, (1998) 507-528 (with D. Rodríguez).

-Complete minimal surfaces derived from Chen-Gackstatter genus two example. Pacific J. Math., Vol. 184, No. 2, (1998) 311-332, (with F. Martín y D. Rodríguez).

-Hyperbolic complete minimal surfaces with arbitrary topology. Trans. Amer. Math. Soc., Vol. 350, May (1998), 1977-1990.

-A characterization of Riemann's minimal examples. J. Diff. Geom., Vol. 47 (1997), 376-397 (with M. Ritoré y F. Wei).

-Complete nonorientable minimal surfaces with the highest symmetry group. Amer. J. Math., Vol. 119, No. 1, February (1997), 55-81 (with F. Martín).

-On complete nonorientable minimal surfaces with low total curvature. Trans. Amer. Math. Soc., Vol. 348, No. 7, July (1996), 2737-2758.

-Complete nonorientable minimal surfaces and symmetries. Duke Math. Journal, Vol. 79 (1995), No. 3, 667-686 (with F. Martin).

-A complete minimal Klein bottle in R3. Duke Math. Journal, Vol. 71, No. 1, (1993) 23-30.

-The classification of complete minimal surfaces with total curvature greater than -12π. Trans. Amer. Math. Soc., Vol. 334 (1992), No. 1, 49-74.

-New complete genus zero minimal surfaces with embedded parallel ends. Proc. Am. Math. Soc., Vol. 112, No. 2, June (1991).

-On embedded complete minimal surfaces of genus zero. J. Diff. Geom., Vol. 33 , No. 1 (1991), 293-300. (with A. Ros).

-Superficies minimales, superficies con curvatura media constante. Ph. D. Thesis, Universidad de Granada (1989).

-Complete minimal surfaces with index one and stable constant mean curvature surfaces. Comm. Math. Helv. , Vol. 64 (1989), 34-43 (with A. Ros).

-A nonorientable complete minimal surface in R3 between two parallel planes. Proc. Am. Math. Soc., Vol. 103, No. 3, July (1988), 913-917.