Global Properties of Biconservative Surfaces in $\mathbb{R}^3$ and $\mathbb{S}^3$

Abstract

Abstract. We survey some recent results on biconservative surfaces in 3-dimensional space forms N 3 (c) with a special emphasis on the c = 0 and c = 1 cases. We study the local and global properties of such surfaces, from extrinsic and intrinsic point of view. We obtain all non-CM C complete biconservative surfaces in R 3 and S 3 .

On the uniqueness of complete biconservative surfaces in $\mathbb {R}^3$

“…Now, if we replace (K ) 2 in (2.9), one gets the following. If ϕ is minimal, then K has to satisfy a fourth order polynomial equation with constant coefficients, with the leading term 4K 4 , or, if ϕ is CM C, we obtain that K has to satisfy a 16-th order polynomial equation with constant coefficients, with the leading term 256K 16 . In both situations, we come to the conclusion that K has to be a constant, and this is a contradiction.…”

“…An important problem is the study of global properties of non-CM C biconservative hypersurfaces in space forms. We note that some global and uniqueness results concerning biconservative surfaces and R 3 and S 3 are given in [13,15,16].…”

Complete biconservative surfaces in the hyperbolic space $\mathbb{H}^3$

“…Theorem 3.1 ( [14,17]). Let M 2 , g(u, v) = e 2σ(u) du 2 + dv 2 be an abstract surface, where u = u(σ) is given by…”

“…We mention that, when the ambient space is R 3 , the result in [8] was reobtained in [2]. Also, some global and uniqueness results concerning biconservative surfaces in R 3 and S 3 are given in [14][15][16][17].…”

“…Then, in Section 3, working in an intrinsic way, we first construct a certain simply connected, complete abstract surface gluing by symmetry along their common boundary two abstract standard biconservative surfaces (see Theorem 3.6). These abstract standard biconservative surfaces were first determined in [17] but, in order to perform the gluing process, we change the coordinates to write the metric in the most appropriate form for our purpose. Then, we prove (in Theorem 3.10), that the above simply connected, complete abstract surface admits a unique biconservative immersion in H 3 .…”