Antonio Martínez

Research topics

• Affine Differential Geometry
• Surfaces of constant Gauss curvature
• Surfaces in Homogeneous Spaces
• Surfaces of non positive curvature
• Geometric PDEs.
• f-minimal surfaces.

Research Grants

• Teoría de superficies y ecuaciones en derivadas parciales geométricas.   Ministerio de Ciencia e Innovación (MICINN) PID2020-118137GB-I00
• Superficies y Ecuaciones en Derivadas Parciales Geométricas.  Ministerio de Economía y Competitividad MTM2016-80313-P
• Superficies y Análisis Geométrico. Junta de Andalucía, Unión Europea P18-FR-4049
• Problemas variacionales y EDPs elípticas en Geometría. Junta de Andalucía A-FQM-139-UGR18
• Problemas Variacionales en Geometría. Junta de Andalucía, FQM-325.
• International Scientific Coordination Network: Geometric Analysis.
• Red Española de Análisis Geométrico. Ministerio de Innovación y Ciencia, MTM2010-09693-E.
• Programa Hispano-Brasileño de Cooperación Interuniversitaria. HPB- 2010-0109
  

Papers

• Geometry of [φ,e_3]-minimal surfaces in R3. arXiv:2205.09467 (with A.L. Martínez Triviño)
• A Weierstrass type representation for translating solitons and singular minimal surfaces. J. Math. Anal. Appl. 516 (2022), no. 2, Paper No. 126528. arXiv:2201.01192 (with A.L. Martínez-Triviño)
• Mean convex properly embedded [φ,e_3]-minimal surfaces in R3. Rev. Mat. Iberoam. 38 (2022), no. 4, 1349-1370. arXiv:2011.15029 (with A.L. Martínez Triviño and Joao P. dos Santos)
• Equilibrium of Surfaces in a Vertical Force Field. Mediterr. J. Math. 19 (2022), no. 1, Paper No. 3, 28 pp. arXiv:1910.07795 (with A.L. Martínez Triviño)
• A Calabi's type correspondence. Nonlinear Anal. 191 (2020), 111637, 17 pp. (with A.L. Martínez Triviño)
  
• Improper affine spheres and the Hessian one equation. Differential Geom. Appl. 54 (2017), part A, 81–90. (with F. Milán)
  
• The Geometric Cauchy Problem for the Hyperbolic Hessian One Equation, Nonlinear Anal (2015). DOI: 10.1016/j.na.2015.05.021 (with F. Milán)
• Complete Surfaces with Ends of Non Positive Curvature, Advances in Mathematics (2015). DOI: 10.1016/j.aim.2015.06.003  (with J. A. Gálvez and J. L. Teruel)
• Complete surfaces with non positive extrinsic curvature in H^3 and S^3, J. Math. Anal. Appl., 430 (2015) pp. 1058-1064, DOI 10.1016/j.jmaa.2015.05.049 (with J. A. Gálvez and J. L. Teruel)
  
• A connection between flat fronts in hyperbolic space and minimal surfaces in euclidean space, Ann. Global Anal. Geom. DOI 10.1007/s10455-015-9468-y, Publ. online 20 May 2015 (with P. Roitman and K. Tenenblat)
  
• A class of surfaces related to a problem posed by Élie Cartan, Annali di Matematica Pura ed Applicata  (4) 195 (2016)no.2, 513-527. (with P. Roitman)
• Ribaucour type transformations for the Hessian one equation. Nonlinear Anal. 112 (2015), 147–155, (with F. Milán and K. Tenenblat)
• Flat fronts in hyperbolic 3-space with prescribed singularities,  Ann. Global Anal. Geom. 46 (2014), no. 3, 227–239,  (with F. Milán)
• Ribaucour transformations for flat surfaces in the hyperbolic 3-space. J. Math. Anal. Appl. 412 (2014), no. 2, 720–743, (with A.V. Corro and K. Tenenblat).
• On the Generalized Weyl Problem for Flat Metrics in the Hyperbolic 3-space, J. Math. Anal. Appl. 410 (2014), no. 1, 144–150. (with J. A. Gálvez and J. L. Teruel )
• Helicoidal Flat Surfaces in the Hyperbolic 3-space,  Pacific J. Math. 264 (2013), no. 1, 195–211, (with J. P. dos Santos and K. Tenenblat)
• Affine theory of plane curves. (Spanish) Gac. R. Soc. Mat. Esp. 16 (2013), no. 2, 271–281. (with F. Milán, and I. Val)
• Contributions in affine differential geometry.   (Spanish) Florentino García Santos: in memoriam, 119-127, Editorial Universidad de Granada, Granada 2011 (with F. Milán)
• An extension of the affine Bernstein problem.  Results Math.,  60 (2011), no. 1-4, 157-174 (with J.A. Aledo and F. Milán).
• Complete flat surfaces with two isolated singularities in hyperbolic 3-space.  J. Math. Anal. Appl. ,  366 (2010), no. 2, 582-592 (with A.V. Corro and F. Milán).
• Affine maximal surfaces with singularities.  Results Math. ,  56 (2009), no. 1-4, 91-107 (with J. A. Aledo and F. Milán).
• Non-Removable Singularities of a fourth-orther nonlinear  partial differential equation.  J. Diff. Equations,  247 (2009), 331-343 (with J.A. Aledo and F. Milán).
• The Affine Cauchy Problem.  J. Math. Anal. Appl. 351 (2009), 70-83 (with J.A. Aledo  and F. Milán).
• The Bonnet Problem for Surfaces in Homogeneous 3-manifolds.  Commun. Anal. Geom. 18 (2008), 907-935 (with J.A. Gálvez  and P. Mira).
• The space of solutions to the Hessian one equation in the finitely punctured plane.  J. Math. Pures Appl. 84 (2005), 1744-1757 (with J.A. Gálvez  and P. Mira).
• A Class of surfaces with flat equiaffine metric and their characterization. Ann. Global Anal. Geom., 28 (2005), 35-57  (with  F. Milán and L. Vrancken.)
• Improper affine maps. Math. Z. 249  (2005),  755-766.
• Complete Linear Weingarten surfaces of Bryan Type. A Plateau Problem at Infinity. Trans. Am. Math. Soc. 356 (2004), no. 9, 3405-3428. (with J. A. Gálvez and F. Milán).
• Complete constant Gauss curvature surfaces in Minkowski space and harmonic diffeomorphisms onto the hyperbolic plane. Tohoku Math. J. 55 (2003), 467-476. (with J.A. Gálvez, and F. Milán)
• Linear Weingarten surfaces in R^3. Monatsh. Math. 138 (2003), 133-144.  (with J.A. Gálvez and F. Milán.)
• The geometry of convex affine maximal graphs. Math. Nachr., 232 (2001) 29-37. (with J. A. Gálvez and F. Milán.)
• Flat surfaces in L^4. Ann. Global anal. Geom.,. 20 (2001), 243-251. (with J. A. Gálvez and F. Milán.)
• Flat surfaces in the hyperbolic 3-space. Math. Ann., 316 (2000), no. 3, 419-435 . (with J. A. Gálvez and F. Milán.)
• The Gauss map and second fundamental form of surfaces in R^3. Geom. Dedicata, 81, (2000), no. 1--3, 181-192. (with J. A. Gálvez.)
• Estimates in surfaces with positive consant Gauss curvature. Proc. Amer. Math. Soc., 128, (2000), no. 12,36553660. (with J. A. Gálvez.)
• The space of parabolic affine spheres with fixed compact boundary. Monatsh. Math. 130, (2000), no. 1, 19-27. (with L. Ferrer and F. Milán ,)
• An extension of a theorem by K. Jörgens and a maximum principle at infinity for parabolic affine spheres, Math. Z. 230, (1999), no. 3, 471-486. (with L. Ferrer and F. Milán,)
• Symmetry and uniqueness of parabolic affine spheres, Math. Ann. 305 , (1996), no. 2, 311-327. ( with L. Ferrer and F. Milán )
• A characterization of the complex paraboloid. Results Math., 27 (1995), no. 3-4, 302-307. ( with F. Milán,)
• Affine isoperimetric problems and surfaces with constant affine mean curvature, Manuscripta Math., 75, (1992), no. 1, 35-41. ( with F. Milán,)
• Recent results on the geometry of real hypersurfaces in quaternionic projective space, Papers in honor of Pablo Bobillo Guerrero (Spanish), 183-190, Univ. Granada, 1992. ( with J.D. Pérez and F.G. Santos,)
• On affine-maximal ruled surfaces, Math. Z., 208, (1991), no. 4, 635-644. ( with F. Milán,)
• On the affine Bernstein problem, Geom. Dedicata., 37, (1991), no. 3, 295-302. ( with F. Milán,)
• On the Pick invariant, the affine mean curvature and the Gauss curvature of affine surfaces, Results Math.20 (1991), no. 3-4, 622-642. (with F. Dillen, F. Milán, F. G. Santos and L. Vrancken,)
• A characterization of the hermitian symmetric spaces of rank three, Bull. Inst. Math. Acad. Sinica, , 17, (1989), no. 2, 157-164.
• Ruled real hypersurfaces in quaternionic projective space, An. \c Stiin\c t. Univ. Al. I. Cuza Ia\c si Sec\c t. I a Mat. 34 (1988), no. 1, 73--78. 
• Real hypersurfaces in quaternionic projective space, Ann. Mat. Pura Appl., 145 (1986), no. 4, 355-384. (with J. D. Pérez.)
• On real hypersurfaces of finite type of CP^m. Kodai Math. J. 7 (1984), no. 3, 304—316. (with A. Ros.)
• Generic submanifolds of a quaternion Kaehlerian manifold. Soochow J. Math., 10, (1984), 79--98. (with J. D. Pérez and F.G. Santos.)
• On the normal connection of quaternion CR-submanifolds. Bull. Inst. Math. Acad. Sinica 12 (1984), no. 3, 237--247. (with J. D. Pérez and F.G. Santos.)

Other publications

• Some geometric aspects of the Hessian one equation, Springer Proceedings in Mathematics, vol. 106, Real and Complex submanifolds, Daejeon, Korea, 2014, 155--165.
• About the Bonnet problem in homogeneous 3-sapces,  proceedings of the symposium on differential geometry of submanifolds. Université de Valenciennes, (2007), 147—154 (with J.A. Gálvez and P. Mira)
• Relatives of flat surfaces in H^3.  proceedings of  the International Workshop on integrable systems, geometry and visualization, Kyushu Univ. Fukuoka, Japan, 115-132, (2005).
• Some results of projectively flat affine surfaces. PDEs, submanifolds, and Affine differential geometry. Banach Center Publications, vol. 69  Institute of Math. Polish Acad. Sciences. Warszaw 2005. (with  F. Milán.)
• Hypersurfaces with constant curvature in R^(n+1). PDEs, submanifolds and affine differential geometry, 101-108, Banach Center Publications, vol. 57, 2002. ( with J. A. Gálvez.) pdf.
• Contact holomorphic curves and flat surfaces, Geometry and topology of submanifolds, X (Beijing/Berlin 1999), 54-61, World Sci. Publishing, River Edge, NJ, 2000. ( with J. A. Gálvez and F. Milán.) pdf
• Improper affine hyperspheres, Geometry and topology of submanifolds, VIII, (Brussels, 1995/Nordfjordeid, 1995), 187-194, World Sci. Publishing, River Edge NJ 1996. ( with L. Ferrer and F. Milán.)
• Affine definite 2-spheres in R^4, Geometry and topology of submanifolds, VII, (Leuven 1994, Brussels, 1994), 182-185, World Sci. Publishing, River Edge NJ 1995. ( with F. Milán.)
• Convex affine surfaces with constant affine mean curvature, Global differential geometry and global analysis (Berlin, 1990) Lecture Notes in Math. 1481, Springer Berlin, 1991. (with F. Milán.)
• Characterizations of locally strongly convex homogeneous affine surfaces, Geometry and topology of submanifolds, V (Leuven/Brussels 1992), 54-61, World Sci. Publishing, River Edge, NJ, 1993. (with F. Milán.)
• Totally complex submanifolds of quaternionic projective space, Geometry and topology of submanifolds (Marseille, 1987), 157--164, World Sci. Publishing, Teaneck, NJ, 1989.
• On the Spectrum of real hypersurfaces of quaternionic projective space. Proceedings of the Eleventh Spanish-Portuguese conference on mathematics, V (Badajoz, 1986), 252--259, Univ. Extremadura, Badajoz, 1987.
• On the second fundamental form of an immersion in a quaternion Kähler manifold. Proceedings of the tenth Spanish-Portuguese conference on mathematics, V (Murcia, 1985), 59--66, Univ. Murcia, Murcia, 1985.