Abstract : The study of minimal surfaces has been many years and reached a much better understanding. Now, it extends to constant mean curvature(CMC) surfaces and to the problems of isometric immersion in higher dimensions or hyperbolic space(Hⁿ), and etc. Here, we try to work on another direction. Consider the condition:

In 2001, Meeks and Rosenberg has the following theorem:

Theorem [Meeks and Rosenberg]. A properly embedded simply-connected minimal surface in R³ is either a plane or a helicoid.

We prove the theorem and it will apply the finite total curvature case of Meeks and Rosenberg's theorem.

Theorem 1. A complete simply-connected embedded C²-surface M in R³ with K £ 0 and (1) is a plane.

If we change the nonpositive curvature condition to negative and move out the simply-connectedness, then we have the following conjecture which relates to some minimal surfaces theorems:

Conjecture 2. The only topology of a complete embedded C²-surface M in R³ with K < 0 and (1) is c(M) = 0 which means M is homeomorphic to an annulus.

Abstract : Let (M, g_M) and (N,g_N) be two smooth oriented Riemannian Manifolds of dimension m and n respectively. Let f: M ® N be a smooth mapping. Then the inverse image f^*g_N is positive semi-definite over M and its trace with respect to g_M, say Trf^*g_N, is a smooth function on M of non-negative values. Let D be a compact domain in M, the integral

When f is an isometric immersion, Chern-Goldberg [] proved that the tension field of f is nothing but the mean curvature vector field of f, and got the following theorem

Theorem 1 Let (M, g_M) and (N, g_N) be two smooth oriented Riemannian Manifolds of dimension m and n respectively. Let f: (M, f^*g_N)® (N, g_N) be an isometric immersion. Then f is harmonic if and only if f is minimal.

Therefore the harmonicity of mappings is a kind of extension of the minimality of immersions.

Definition 1 Let f: (M, g_M)® (N, g_N) be a smooth mapping. We define the graph of f to be the mapping F: M® M×N, F(x) = (x, f(x)), for all x Î M.

It is clear that F is an isometric embedding of M into (M×N, g_MÅg_N) as a closed submanifold under the induced metric g_M+f^*g_N.

When M = S², RP² and N = S², Eells [] proved that the graph of harmonic mapping f are minimal. Note that in these cases, M is obviously a conformal mapping.

Schoen [] studies the Bernstein typed problem of smooth mapping f and proved the following

Theorem 2 Let M be a 2-dimensional complete Riemannian manifold of non-negative curvature which is not flat and N be an arbitrary Riemannian manifold. Suppose that f: M®N be a minimal mapping (i.e. its graph is a minimal embedding). Then f is necessarily a conformal harmonic mapping. Conversely, any conformal harmonic mapping f: M® N must be a minimal mapping.

The aim of the present paper is to study the graph of a smooth mapping systematically. We first give expression of the mean curvature vector field of its graph (see Proposition 1). Then we prove the following main result:

Theorem Let (M, g_M) and (N, g_N) be two smooth oriented Riemannian Manifolds of dimension m and n respectively. Let f: (M, g_M)® (N, g_N) be a smooth mapping. If m > 2, then the graph of f,

This result generalized Theorems 1 and 2.

Reference

[1] S. S. Chern and S. I. Goldberg, On the volume decreasing property of a class of real harmonic mappings, Amer. J. Math. 97(1975), No. 1, 133-147.

[2] Eells, Jr., Minimal graphs, Manus. Math. 28(1979), 101-108.

[3] R. Schoen, The Role of Harmonic Mappings in Rigidity and Deformation Problems, Complex Geometry, Lecture Notes in Pure and Applied Math\. 143, Dekker, New York, 1993, 179-200.

Abstract : In this talk, I plan to explain some results concerning the space of polynomial growth harmonic forms. We proved that the dimension of such spaces must be finite and can be estimated if the metric is uniformly equivalent to one with nonnegative curvature operator. In particular, this implies that the space of harmonic forms of fixed growth order on the Euclidean space with any periodic metric must be finite dimensional.

Abstract : In this paper we use Gauss map to study spacelike submanifolds in de Sitter space form. We prove that if there exist r > 0 and a fixed unit simple (n + 1)-vector a Î G^p_{n+1, p} such that the Gauss map g of an n-dimensional complete and connected spacelike submanifold Mⁿ in S^n+p satisfies ág, a ñ £ r, then Mⁿ is diffeomorphic to Sⁿ, and its volume satisfies vol(Sⁿ)/r £ vol(M) £ rⁿvol(Sⁿ). We also characterize the case where these inequalities become equalities.

Abstract : Let f:\mathbbR²® \mathbb R² be an area preserving diffeomorphism between R². The graph of f can be viewed as Lagrangian submanifold in R⁴. In this talk, we show that the long time existence for the mean curvature flow of an area preserving map between \mathbb R² .

Abstract : Consider the planar parametric curves based on the basis 1, t, j(t) and y(t) as follows:

In this paper, we analyze some important geometric properties of the curves r(t), including the distribution of cusps, loops, inflection points on r(t), and necessary and sufficient conditions for those curves containing the above points in terms of their control polygons.

We prove that: Suppose p₂ is not parallel to p₃, p₁ = lp₂+mp₃; d(t) = (j¢(t),y¢(t))^T (0 £ t £ 1) is a planar convex curve; T₀ and T₁ are the tangent lines of d(t) at its two end points, respectively, such that their intersection lies in the convex side of d(t), and none of them intersects with d(t) at inner points; g_d(t) = det(d¢(t),d¢¢(t)) does not change its sign and d¢(t) ¹ 0 when 0 £ t £ 1. Then, geometric properties of r(t) are all determined by the position of the point in lm-plane as follows

1. r(t) has a cusp if and only if (l,m) Î C = {-d(t):0 £ t £ 1};

2. r(t) has an inflection point, if and only if (l,m) Î S, where S is composed of two domains surrounded by T₀ and T₁, and each of the domains and C belong to the same side of T₀ and T₁, respectively;

3. r(t) has two inflection points, if and only if (l,m) Î D, where D is surrounded by T₀, T₁, and C, where C Ï D;

4. r(t) has a loop if and only if (l,m) Î L, which is surrounded by L₁, L₂, and C with C Ï L, L₁ = {(l,m):l = -[0,t]j,m = -[0,t]y,0 < t £ 1}, L₂ = {(l,m):l = -[t,1]j,m = -[t,1]y,0 £ t < 1} , where [a,b]f denotes the difference quotient of f with knots a and b;

5. r(t) has none of cusp, loop and inflection point, if and only if (l,m) Î N, where N is the complement of C, S and D in lm-plane.

The above conclusion contains the known results for the curves as follows:

1. r(t) is the general C-curve with j(t) = sin(at),y(t) = cos(at);

2. r(t) is the parametric curve under tension with j(t) = sinh(r(1-t)),y(t) = cosh(rt);

3. r(t) is the parametric curve of order 4 with power function basis, where j(t) = (1-t)^a,y(t) = t^b; and specially, r(t) is the planar parametric polynomial curve of degree 3 when a = b = 3 .