Hsungrow Chan
Department of Mathematics Education
National Pingtung Teachers College
Pingtung, Taiwan
hchan@mail.npttc.edu.tw
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(H^{n}), and etc. Here, we try to work on another direction. Consider the condition:
 (1) 
In 2001, Meeks and Rosenberg has the following theorem:
Theorem [Meeks and Rosenberg]. A properly embedded simplyconnected minimal surface in R^{3} 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 simplyconnected embedded C^{2}surface M in R^{3} with K £ 0 and (1) is a plane.
If we change the nonpositive curvature condition to negative and move out the simplyconnectedness, then we have the following conjecture which relates to some minimal surfaces theorems:
Conjecture 2. The only topology of a complete embedded C^{2}surface M in R^{3} 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 semidefinite over M and its trace with respect to g_{M}, say Trf^{*}g_{N}, is a smooth function on M of nonnegative values. Let D be a compact domain in M, the integral

When f is an isometric immersion, ChernGoldberg [] 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_{M}Åg_{N}) as a closed submanifold under the induced metric g_{M}+f^{*}g_{N}.
When M = S^{2}, RP^{2} and N = S^{2}, 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 2dimensional complete Riemannian manifold of nonnegative 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, 133147.
[2] Eells, Jr., Minimal graphs, Manus. Math. 28(1979), 101108.
[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, 179200.
JuiTang Chen
Department of Mathematics
National Chung Cheng University
Minhsiung, Chiayi Taiwan 62117
d8921001@mthmp.math.ccu.edu.tw
ChiungJue Anna Sung^{*}
Department of Mathematics
National Chung Cheng University
Minhsiung, Chiayi Taiwan 62117
cjsung@math.ccu.edu.tw
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.
BingYe Wu
Institute of Mathematics
Fudan University
Shanghai, P.R. China
wubingye@zjnu.cn
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 ndimensional complete and connected spacelike submanifold M^{n} in S^{n+p} satisfies ág, a ñ £ r, then M^{n} is diffeomorphic to S^{n}, and its volume satisfies vol(S^{n})/r £ vol(M) £ r^{n}vol(S^{n}). We also characterize the case where these inequalities become equalities.
ShuCheng Chang
Department of Mathematics
National Tsing Hua University
Hsinchu, Taiwan 30013, R.O.C.
scchang@math.nthu.edu.tw
MaoPei Tsui
Department of Mathematics
Columbia University
New York, NY 10027, USA
tsui@math.columbia.edu
ChinTung Wu^{*}
Department of Mathematics Education
National Pingtung Teachers College
Pingtung, Taiwan 90003, R.O.C.
ctwu@mail.npttc.edu.tw
Abstract : Let f:\mathbbR^{2}® \mathbb R^{2} be an area preserving diffeomorphism between R^{2}. The graph of f can be viewed as Lagrangian submanifold in R^{4}. In this talk, we show that the long time existence for the mean curvature flow of an area preserving map between \mathbb R^{2} .
Zhenglin Ye^{*} and Rongjun Wu
Department of Applied Mathematics
Northwestern Polytechnical University
Xi'an, Shaanxi 710072, P.R. China
yezhenglin@sina.com
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 p_{2} is not parallel to p_{3}, p_{1} = lp_{2}+mp_{3}; d(t) = (j¢(t),y¢(t))^{T} (0 £ t £ 1) is a planar convex curve; T_{0} and T_{1} 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 lmplane as follows
The above conclusion contains the known results for the curves as follows: