Study of the structure of hypersurfaces with constant scalar curvature

在欧氏空间中，已知的具有正常数曲率的完全超曲面只有球面、广义柱面S^p×R^<n-p>和旋转超曲面。本文在欧几里德空间中构造了无穷多个具有常正标量曲率的完全超曲面的新例子。对于O（p+1）×O(q+1)-不变超曲面，表示其标量曲率的方程为常数S为(I){dx/ds=cosα, dy/ds=sinα, dα/ds=(p(p-1)((sinα)/x)^2-2pq(sinα)/x(cosα)/y)^2-S)/(2(q(cosα)/y)- p(sinα)/x))，其中α是切向量（x', y'）与x轴之间的夹角。本研究的重点是我们可以将解与另一个显积ODE的解进行比较。定理设p <小于等于q + 1， S > （p <大于等于2）。令0 < x_0【小于或等于】√<p(p -1)/S>且0 < y_0【小于或等于】√<q(q -1)/S>。那么对于初始条件x(0) = x_0, y(0) = y_0, α(0) = 0， ODE系统在（-∞,∞）上有一个全局解γ(s) = (x(s), y(s))∈R_+ × R_+。因此，M_γ在R^<p+q+2>中成为具有恒定标量曲率S的完全超曲面。定理设p > q+ 1和S > 0 （q【大于等于】2）。令0 < x_0【小于等于】√<(p -1)(q -1)/S>且0 < y_0【小于等于】√<q(q -1)/S>。那么对于初始条件x(0) = x_0, y(0) = y_0, α(0) = 0， ODE系统在（-∞,∞）上有一个全局解γ(s) = (x(s), y(s))∈R_+ × R_+。因此，M_γ在R^<p+q+2>中成为具有常数曲率S的完全超曲面。

In the Euclidean spaces the known examples of complete hypersurfaces with positive constant scalar curvature are only spheres, generalized cylinders S^p×R^<n-p> and the rotational hypersurfaces. In this research we constructed infinitely many new examples of complete hypersurfaces with constant positive scalar curvature in the Euclidean spaces.For the O(p+1)×O(q+1)-invariant hypersurfaces, the equation representing its scalar curvature is constant S is(I){dx/ds=cosα, dy/ds=sinα, dα/ds=(p(p-1)((sinα)/x)^2-2pq(sinα)/x(cosα)/y+q(q-1)((cosα)/y)^2-S)/(2(q(cosα)/y-p(sinα)/x)),where α is the angle between the tangent vector (x', y') and the x-axis.The key point of this study is that we can compare the solution with another ODE which is explicitly integrable.Theorem Suppose that p 【less than or equal】 q + 1 and S > 0 (p 【greater than or equal】 2). Let 0 < x_0 【less than or equal】 √<p(p -1)/S> and 0 < y_0 【less than or equal】 √<q(q -1)/S>. Then the ODE system has a global solution γ(s) = (x(s), y(s)) ∈ R_+ × R_+ on (-∞, ∞) for the initial condition x(0) = x_0, y(0) = y_0 and α(0) = 0. Therefore M_γ become a complete hypersurface in R^<p+q+2> with constant scalar curvature S.Theorem Suppose that p > q + 1 and S > 0 (q 【greater than or equal】 2). Let 0 < x_0 【less than or equal】 √<(p -1)(q -1)/S> and 0 < y_0 【less than or equal】 √<q(q -1)/S>. Then the ODE system has a global solution γ(s) = (x(s), y(s)) ∈ R_+ × R_+ on (-∞, ∞) for the initial condition x(0) = x_0, y(0) = y_0 and α(0) = 0. Therefore M_γ become a complete hypersurface in R^<p+q+2> with constant scalar curvature S.