Let s ： S2 → G（2, 5） be a linearly full totally unramified pseudo-holomorphic curve with constant Gaussian curvature K in a complex Grassmann manifold G（2, 5）. It is prove that K is either 1 4 1 or 4/5 if s is...Let s ： S2 → G（2, 5） be a linearly full totally unramified pseudo-holomorphic curve with constant Gaussian curvature K in a complex Grassmann manifold G（2, 5）. It is prove that K is either 1 4 1 or 4/5 if s is non-±holomorphic. Furthermore, K = 1/3 if and only if s is totally real. We also prove that the Gaussian curvature K is either 1 or -4/3 if s is a non-degenerate holomorphic curve under some conditions.展开更多
For an almost complex structure J on U R4 pseudo-holomorphically fibered over C a J-holomorphic curve C U can be described by a Weierstrass polynomial. The J-holomorphicity equation descends to a perturbed 8-operator ...For an almost complex structure J on U R4 pseudo-holomorphically fibered over C a J-holomorphic curve C U can be described by a Weierstrass polynomial. The J-holomorphicity equation descends to a perturbed 8-operator on the coefficients; the operator is typically (0, 2/m)-Holder continuous if m is the local degree of C over C. This sheds some light on the problem of parametrizing pseudo-holomorphic deformations of J-holomorphic curve singu-larities.展开更多
Further geometry and topology for pseudo-holomorphic curves in complex Grassmannians Gm(CN) are studied. Some curvature pinching theorems for pseudo- holomorphic curyes with constant Kahler angles in Gm(CN) are obtain...Further geometry and topology for pseudo-holomorphic curves in complex Grassmannians Gm(CN) are studied. Some curvature pinching theorems for pseudo- holomorphic curyes with constant Kahler angles in Gm(CN) are obtained, so that the corresponding results for pseudoholomorphic curves in complex projective spaces are generalized.展开更多
In this paper, we study the Hofer-Zehnder capacity and the Weinstein conjecture in symplectic manifold (M×R^(2n), ω(?)σ). Let us define l_1(M, ω)=inf{<ω, α>|>0, α∈π_2(M)}. Suppose l_1(M, ω)>O...In this paper, we study the Hofer-Zehnder capacity and the Weinstein conjecture in symplectic manifold (M×R^(2n), ω(?)σ). Let us define l_1(M, ω)=inf{<ω, α>|>0, α∈π_2(M)}. Suppose l_1(M, ω)>O, O<πr^2<2/1 l_1(M, ω). Then C_(HZ)(M×B(r))=C_(HZ)(M×Z(r))=πr^2. In the case M is a point {P}, we obtain the well-known result at present. For n>1, consider on Cp^(n-1) the standard symplectic form co such that ω[u]=n for a generator u of H_2(CP^(n-1). Suppose O<πr^2<2/1 n. ThenC_(HZ)(M×B(r))=C_(HZ)(M×Z(r))=πr^2.As an application, we claim that the Weinstein conjecture in M×Z(r) is proved correct.展开更多
基金Supported by National Natural Science Foundation of China (Grant No. 10531090)Knowledge Innovation Funds of CAS (KJCX3-SYW-S03)+1 种基金 SRF for ROCS,SEMthe President Fund of GUCAS
文摘Let s ： S2 → G（2, 5） be a linearly full totally unramified pseudo-holomorphic curve with constant Gaussian curvature K in a complex Grassmann manifold G（2, 5）. It is prove that K is either 1 4 1 or 4/5 if s is non-±holomorphic. Furthermore, K = 1/3 if and only if s is totally real. We also prove that the Gaussian curvature K is either 1 or -4/3 if s is a non-degenerate holomorphic curve under some conditions.
文摘For an almost complex structure J on U R4 pseudo-holomorphically fibered over C a J-holomorphic curve C U can be described by a Weierstrass polynomial. The J-holomorphicity equation descends to a perturbed 8-operator on the coefficients; the operator is typically (0, 2/m)-Holder continuous if m is the local degree of C over C. This sheds some light on the problem of parametrizing pseudo-holomorphic deformations of J-holomorphic curve singu-larities.
文摘Further geometry and topology for pseudo-holomorphic curves in complex Grassmannians Gm(CN) are studied. Some curvature pinching theorems for pseudo- holomorphic curyes with constant Kahler angles in Gm(CN) are obtained, so that the corresponding results for pseudoholomorphic curves in complex projective spaces are generalized.
基金Project supported by the Science Foundation of Tsinghua University
文摘In this paper, we study the Hofer-Zehnder capacity and the Weinstein conjecture in symplectic manifold (M×R^(2n), ω(?)σ). Let us define l_1(M, ω)=inf{<ω, α>|>0, α∈π_2(M)}. Suppose l_1(M, ω)>O, O<πr^2<2/1 l_1(M, ω). Then C_(HZ)(M×B(r))=C_(HZ)(M×Z(r))=πr^2. In the case M is a point {P}, we obtain the well-known result at present. For n>1, consider on Cp^(n-1) the standard symplectic form co such that ω[u]=n for a generator u of H_2(CP^(n-1). Suppose O<πr^2<2/1 n. ThenC_(HZ)(M×B(r))=C_(HZ)(M×Z(r))=πr^2.As an application, we claim that the Weinstein conjecture in M×Z(r) is proved correct.
