zbMATH — the first resource for mathematics

Globally analytic hypoelliptic vector fields on compact surfaces. (English) Zbl 1139.35038
This paper deals with complex, non-singular, real analytic vector field $$L$$ defined on a compact, connected, orientable, two-dimensional, real analytic manifold $$M$$. The authors call $$L$$ vector field of type I if $$\text{Re\,}L$$, $$\text{Im\,}L$$ are linearly independent everywhere on $$M$$. Otherwise, $$L$$ is called vector field of type II. A complete characterization of the global analytic hypoelliptic vector fields of types I and II is given. It follows from their results that in the class of compact surfaces $$M$$, globally analytic hypoelliptic vector fields can exist only if $$M$$ is real analytically diffeomorphic to the two-torus $$\mathbb{T}^2$$.

MSC:
 35H10 Hypoelliptic equations 58J99 Partial differential equations on manifolds; differential operators
Full Text:
References:
 [1] M. S. Baouendi and F. Trèves, A local constancy principle for the solutions of certain overdetermined systems of first-order linear partial differential equations, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 245 – 262. [2] Adalberto P. Bergamasco, Remarks about global analytic hypoellipticity, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4113 – 4126. · Zbl 0932.35046 [3] Adalberto P. Bergamasco, Wagner V. L. Nunes, and Sérgio Luís Zani, Global properties of a class of overdetermined systems, J. Funct. Anal. 200 (2003), no. 1, 31 – 64. · Zbl 1034.32024 · doi:10.1016/S0022-1236(02)00055-1 · doi.org [4] Adalberto P. Bergamasco and Sérgio Luís Zani, Prescribing analytic singularities for solutions of a class of vector fields on the torus, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4159 – 4174. · Zbl 1077.35004 [5] Paulo Cordaro and Jorge Hounie, On local solvability of underdetermined systems of vector fields, Amer. J. Math. 112 (1990), no. 2, 243 – 270. · Zbl 0708.58025 · doi:10.2307/2374715 · doi.org [6] Hans Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460 – 472. · Zbl 0108.07804 · doi:10.2307/1970257 · doi.org [7] Stephen J. Greenfield, Hypoelliptic vector fields and continued fractions, Proc. Amer. Math. Soc. 31 (1972), 115 – 118. · Zbl 0229.35024 [8] Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. · Zbl 0356.57001 [9] Jorge Hounie, Minimal sets of families of vector fields on compact surfaces, J. Differential Geom. 16 (1981), no. 4, 739 – 744 (1982). · Zbl 0471.58019 [10] J. Hounie, Globally hypoelliptic vector fields on compact surfaces, Comm. Partial Differential Equations 7 (1982), no. 4, 343 – 370. · Zbl 0588.35064 · doi:10.1080/03605308208820226 · doi.org [11] Héctor J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171 – 188. · Zbl 0274.58002 [12] François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. · Zbl 0565.35079
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.