行政院國家科學委員會補助專題研究計畫
▉ 成 果 報 告
□期中進度報告
負曲率曲面漸近線之研究
計畫類別：▉ 個別型計畫 □ 整合型計畫
計畫編號：
NSC97-2115-M-153-001
執行期間：2008/08/01 ~ 2009/07/31
計畫主持人： 詹勳國
Hsungrow Chan hchan@mail.npue.edu.tw這份報告分成4項小節，一學術報告、二參與研討會與討論會、
三發表論文，四學術進展。
一、學術報告
2008 非線性分析及幾何分析研討會 2008 Workshop on Nonlinear Analysis and
Geometric Analysis (under-construction) 時間：2008 年 9 月 5 日（週五）中午至
8 日 Time: Sep. 5-8, 2008
國立中山大學應用數學系 (Dept. of Applied Math., National Sun Yat-sen University)
Talk Speaker：詹勳國 教授 (Hsungrow Chan) Title：黑洞模型與非正高斯曲率曲
面論問題 Time：2008/10/09（Thursday,星期四）15：30 ~ 16：30
2008 AMMS 數學學術研討會暨中華民國數學會年會 報告 詹勳國. Immersed
equations of the Hyperbolic Surfaces. 15：30～15：50 ...
報告題目：Nonpositively Curved Surfaces 報告人：詹勳國教授 2009 1 月 8 日 15：
00-16：00 地點：廣州中山大學 210 室
報告題目：Nonpositively Curved Surfaces 報告人：詹勳國教授 2009：3 月 30 日
10：00-11：00 地點：上海華東師範大學數學系
二、參加研討會與討論會
因為所服務學校沒有 AMS sci-net 與數學藏書，所以經常前往國立清華大學理
論中心借用研究室，與使用數學系圖書館。
July 18 Ni Lei "classification of gradient flow" NTU math.
Mathematics, National Taiwan University, Taipei, Taiwan
成功大學南區理論中心 2008/12/27~2008/12/27 Fujisan One-day Workshop in
Geometry and Topology
成 功 大 學 南 區 理 論 中 心 2009/04/12~2009/04/12 Mini-workshop on Contact
Geometry
2009-06-11 新竹地區數學研討會 Lecture Room A of NCTS 4th Floor, The 3rd
General Building , NTHU
2009 。 7 月開始 台大「數學月」一系列的活動
三、發表論文
2008 年 6 月投稿至 NAGOYA 數學雜誌的回覆意見
審查意見一
The author proves the topological uniqueness of complete surfaces. The proof is a
clever use of the index formula obtained by Verner [12] for asymptotic line field. The
result looks interesting even if the method to prove it is rather classical.
In fact, one can show the following as an application:
Corollary. The only complete embedded minimal surfaces with finite total curvature and
without umbilic points are the plane and the catenoid.
Therefore I would recommend to publish it from Nagoya if the author improved
the exposition substantially. In particular, the author should emphasize the geometric
meanings and applications of the main theorem. The attached comments will be helpful
for the author to revised the manuscript.
審查意見二
The main result of the paper under review is that a complete negatively curved
surface in Euclidean 3-space which has finite total curvature must be an annulus.
undertakes the task of vigorously rewriting and polishing this work, and produces a
more concise and precise manuscript.
綜合意見：只要改善英文寫作，2 位意見都推薦刊登，已經改善回覆。
2008 年 9 月投稿一篇論文至物理雜誌 CHAOS, SOLITONS & FRACTALS。已經
一年仍未回覆任何消息。
四、學術進展
The Hadamard’s Conjecture for embedded surfaces
Hsungrow Chan
Abstract
The Hadamard’s Conjecture claims that a complete negatively curved surface in R3 must be unbounded. In 1996, Nadirashvili constructed a complete immersion of a negatively curved surface into the unit ball in R3, showing the Conjecture failed for immersed surfaces. In this talk, we prove the Hadamard’s Conjecture for embedded surfaces (i.e. surfaces without self-intersection) under the condition of finite total curvature. Also, we will show the progress under the condition of infinite total curvature.
hchan@mail.npue.edu.tw
Hadamard’s Conjecture. A complete negatively curved surface in R3 must be un-bounded.
Calabi-Yau’s Conjecture. A complete minimal surface in R3 must be unbounded.
In 1996, Nadirashvili [2] constructed a complete immersion of a minimal disk into the unit ball in R3, showing the Calabi-Yau conjecture failed for immersed surfaces. The Gauss curvature of his construction is negative. Hadamard’s conjecture also failed for the immersed negatively curved surface. In 2008, Colding and Minicozzi [1] proved that the Calabi-Yau conjecture is true for the embedded surfaces. We believe that the embedded negatively curved surfaces in R3 behave the same:
Theorem 1. A complete embedded negatively curved surface in R3 is unbounded.
The proof is divided into two cases; finite total curvature and infinite total curvature.
1. Finite Total Curvature
We divide the Hadamard’s Conjecture for the embedded surfaces into 2 cases. The first one is the case of finite total curvature. The conjecture is true even for the immersed surface.
Theorem 2. A complete immersed negatively curved surface in R3 with finite total curvature is unbounded.
The proof combines the index method, asymptotic curves, and the hyperbolic partial differential equations. The finite total curvature implies the finite topology. Outside of a big compact set of the surface is the union of finite many ends. Each end has certain sectors of asymptotic curves. By the isometric immersions, the asymptotic sector is unbounded and the end is unbounded. Thus, a complete immersed negatively curved surface in R3
with finite total curvature is unbounded
2. Infinite Total Curvature
Let M be a complete embedded negatively curved surface in R3.
Definition 1. For any p ∈ M , let γ : [0, a] → M be a geodesic where a could be a finite number or infinite. The point γ(t0) is said to be Gauss injective point to γ(0) = p along
γ if the gauss map is 1-1 along γ for 0 ≤ t ≤ t0. We define the gauss injective region of
p by CM(p) as the union of the gauss injective points of p along all of the geodesics that
start from p.
CM(p) is well defined for the negativity implying locally 1-1 gauss map. CM(p) is the
largest p-neighborhood of M such that the gauss map
g : CM(p) → S2
is 1-1.
Let g : M ⊂ R3 → S2 be the gauss map. The embeddedness implies that g(C M(p))
cannot be the whole S2. The negativity implies that g(C
M(p)) cannot be compact in S2.
g(CM(p)) is not close in S2. There is a boundary point of g(CM(p)) which is not included
in g(CM(p)).
References
[1] Colding, T. H. & Minicozzi, W. P., The Calabi-Yau conjectures for embedded surfaces. Ann. Math. 1(2008), 211-243.
[2] Nadirashvili, N. Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces. Inven. Math. 126(1996), 457-465.
國外差旅費出國報告
負曲率曲面漸近線之研究
NSC97-2115-M-153-001
詹勳國 Hsungrow Chan hchan@mail.npue.edu.tw
應用數學系，國立屏東教育大學，屏東市 900-03 民生路 4-18 號 執行計畫期間(一)訪問廣州中山大學數學系陳兵龍教授，(二)訪問上海華東師範大學數學系潘 生亮教授，(三)訪問上海復旦大學數學系忻元龍教授，(四) 訪問上海復旦大學數學系洪家興 教授 (一) 時間：98 年 1 月 4 日至 1 月 10 日，受系主任陳兵龍教授邀請，訪問廣州市中山大學數學與 計算科學學院做短期之學術研究。 訪問的目的之一，是計畫要推廣以下定理，也有多位學者關心與提出這方向的問題。 Theorem 1. A complete simply connected embedded C -surface M in 2 R with 3 K ≤0 and
∫
2<∞ | | B is a plane. 到高維度的情形，因為原來的證明是利用 2 維拓普與幾何量的性質， 而這些一旦提升到 3 維以上，性質不復出現。因此方法是先想求出分析的證明，而陳兵龍教 授熟悉這方面的最適合人選。陳兵龍教授是朱熹平教授與丘成桐教授的學生，曾經訪問台灣 數次，分析的功力很好，工作也很突出，發表的論文如下，都是著名的期刊，質量均佳，是 中國數學界的後起之秀，前途看好。訪問期間很快的陳教授就給一個合理的證明方向，利用 比較定理，因為從遠方來看幾乎是平面，因此面積與擴展半徑的比有上下界，最後會收斂成 一個值，而這個值等同於選內部點時的ε 小區域的比值，所以它必須是一個平面，而這些都 是利用分析手法來作的，可惜的是這個方法同樣的無法推廣到高維度的情形。期間也提到 Spruch 教授的他建議：一是利用 L. Simon 的不等式($|d\Phi|^2\leq
c1\Phi_*+c2$)，給一個純分析的證明，其中的 c2 把它改成下降至零的有上界數，當然要對 PDE 還要 push 多一點點。第二建議是使用 L.Bers 的複幾何方程式$Q_{\bar z}=H_{z}$in weak sense，來處理。 他覺得非常有意思，做的人不多，有一定的困難度，所以願意繼續試試看， 並維持討論。
訪問期間也對廣州中山大學做一個報告：
報告題目：Nonpositively Curved Surfaces 报告人：詹勋国教授 时间：1 月 8 日 15：00-16：00
陳兵龍教授的著作
1. Chen, Bing-Long; Zhu, Xi-Ping Surgical Ricci flow on four-manifolds with positive isotropic curvature. Third International Congress of Chinese Mathematicians. Part 1, 2, 101--109, AMS/IP Stud. Adv. Math., 42, pt.1, 2, Amer. Math. Soc., Providence, RI, 2008. 53C44
2. Chen, Bing-Long; LeFloch, Philippe G. Injectivity radius of Lorentzian manifolds. Comm. Math. Phys. 278 (2008), no. 3, 679--713. (Reviewer: Angel Ferrández) 53C50
3. Chen, Bing-Long; Yin, Le Uniqueness and pseudolocality theorems of the mean curvature flow. Comm. Anal. Geom. 15 (2007), no. 3, 435--490. (Reviewer: Fernando Schwartz) 53C44 (35K55)
4. Chen, Bing-Long; Zhu, Xi-Ping Uniqueness of the Ricci flow on complete noncompact manifolds. J. Differential Geom. 74 (2006), no. 1, 119--154. (Reviewer: Weimin Sheng) 53C44
5. Chen, Bing-Long; Zhu, Xi-Ping Ricci flow with surgery on four-manifolds with positive isotropic curvature. J. Differential Geom. 74 (2006), no. 2, 177--264. (Reviewer: Peng Lu) 53C44 (53C21)
6. Chen, Bing-Long; Fu, Xiao-Yong; Yin, Le; Zhu, Xi-Ping Sharp dimension estimates of holomorphic functions and rigidity. Trans. Amer. Math. Soc. 358 (2006), no. 4, 1435--1454 (electronic). (Reviewer: Peng Lu) 32Q30 (32Q10 32Q15)
7. Chen, Bing-Long; Zhu, Xi-Ping Volume growth and curvature decay of positively curved Kähler manifolds. Q. J. Pure Appl. Math. 1 (2005), no. 1, 68--108. (Reviewer: James McCoy) 53C55 (53C21 53C44)
8. Chen, Bing-Long; Tang, Siu-Hung; Zhu, Xi-Ping A uniformization theorem for complete non-compact Kähler surfaces with positive bisectional curvature. J. Differential Geom. 67 (2004), no. 3, 519--570. (Reviewer: Peng Lu) 53C44 (53C21)
9. Cao, Huai-Dong; Chen, Bing-Long; Zhu, Xi-Ping Ricci flow on compact Kähler manifolds of positive bisectional curvature. C. R. Math. Acad. Sci. Paris 337 (2003), no. 12, 781--784. (Reviewer: Peng Lu) 53C44 (53C55)
10. Chen, Bing-Long; Zhu, Xi-Ping On complete noncompact Kähler manifolds with positive bisectional curvature. Math. Ann. 327 (2003), no. 1, 1--23. (Reviewer: Man Chun Leung) 53C21 (32Q10 53C55)
surfaces. Geometry and nonlinear partial differential equations (Hangzhou, 2001), 5--12, AMS/IP Stud. Adv. Math., 29, Amer. Math. Soc., Providence, RI, 2002. (Reviewer: Peng Lu) 53C44 (32Q15 53C55)
12. Chen, Bing-Long; Zhu, Xi-Ping A gap theorem for complete noncompact manifolds with nonnegative Ricci curvature. Comm. Anal. Geom. 10 (2002), no. 1, 217--239. (Reviewer: Shu-Yu Hsu) 53C21
13. Chen, Bing-Long; Zhu, Xi-Ping The Ricci flow on complete three-manifolds. Progress in nonlinear analysis (Tianjin, 1999), 24--45, Nankai Ser. Pure Appl. Math. Theoret. Phys., 6, World Sci. Publ., River Edge, NJ, 2000. 53C44 (35K55)
14. Chen, Bing Long Type III singularity for mean curvature flows. (Chinese) Acta Sci. Natur. Univ. Sunyatseni 39 (2000), no. 2, 131--132. (Reviewer: Shu-Cheng Chang) 53C44
15. Chen, Bing-Long; Zhu, Xi-Ping Complete Riemannian manifolds with pointwise pinched curvature. Invent. Math. 140 (2000), no. 2, 423--452. (Reviewer: Peng Lu) 53C44 (53C21)
(二)
時間：98 年 3 月 26 日至 4 月 2 日，受潘生亮教授邀請，訪問上海地區的數學相關學者。 2007 年參加南京的幾何分析會議 International Conference in Geometry and Analysis，聽了潘生亮教授 的演講 Shengliang Pan (ECNU, Shanghai) On a new curve evolution problem in the plane，之後也和 他討論，他的條件是在限制面積之下的流，這個問題也被推廣到高維度的情形，因為我也有 類似關於相對論的問題，和他討論，同時他對我的研究工作有興趣，特別是利用電腦軟體 MAPLE 的輔助，當時他有邀請到上海華東師範大學給演講與繼續討論。因此 2009 年安排上 海行，同時也拜訪一直利用電子郵件交流的上海復旦大學數學系忻元龍教授，與熟識多年的 上海復旦大學數學洪家興教授。 和潘教授討論的議題是來自於相對論，經過化簡後的宇宙模型經常是R× 拓普形式，S1 若是可以嵌入 3 R ，一個簡單的情形可以想像成類似正在旋轉中的旋轉曲面(曲率可以正可以 負)，這時必須考慮時間的演化，一般來說最穩定的狀態就是圓柱面，所以要找到一個合理的 模型，來描述這個情形，但是又不能隨著時間流到一條線，一開始曲面至主軸有一個最短距 離的點，在旋轉演化中，這個最短距離必須限制住﹝潘教授的工作是：限制面積之下的流﹞， 而最後流至最穩定的狀態就是圓柱面的半徑就必須是一開始曲面至主軸有一個最短距離，而 這必須被推廣至高維度的情形。
spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. 該 論文主要針對閉的曲面，而我們要討論的是開而完備的曲面，但是所用的模型是相似的，因 為在特定條件下，是切下奇異點鄰域後會流到cylindrical，這剛好可以解釋相對論的想法，所 以，合理的朝這方向修改，應可以完成這問題。
另外，若是再簡化成一維嵌入二維，那就是完備非閉曲線接近一直線有最近點，如何在 流的過程中，保持這距離，然後流到一直線2007年6月13日 arXiv:0705.3827 Tobias H. Colding, William P. Minicozzi II 有一篇 Width and mean curvature flow 論文可以參考。所以mean curvature flow和相對論的問題有一定的關聯，可以持續研究。
訪問期間應邀做一個學術報告：
報告題目：Nonpositively Curved Surfaces 报告人：詹勋国教授 时间：3 月 30 日 10：00-11：00
地点：上海華東師範大學数学系
潘生亮 2001 年以来发表的论文:
[1] S. L. Pan & H. P. Xu‚ Stability of a reverse isoperimetric inequality‚ J. Math. Anal. Appl. ‚ 350(2009)‚ 348-353.
[2] S. L. Pan & J. N. Yang‚ On a nonlocal perimeter-preserving curve evolution problem for convex plane curves‚ Manuscrupta Math.‚ 127(2008)‚ 469-484.
[3] 潘生亮、唐学远、汪小玉 Gage 等周不等式的加强形式数学年刊 29A（2008） 301-306.
[4] S. L. Pan‚ X‚ W‚ Sun & Y. D. Wang‚ A variant of isoperimetric problem‚ I‚ Science in China Series A: Mathematics 51:6(2008)‚ 1119-1126.
[5] L. S. Jiang & S. L. Pan‚ On a non-local curve evolution problem in the plane‚ Communications in Analysis and Geometry‚ 16:1(2008)‚ 1-26.
[6] S. L. Pan & H. Zhang‚ A reverse isoperimetric inequality for closed strictly convex plane curves‚ Beiträge zur Algebra und Geoemtrie‚ 48:1 (2007)‚ 303-308.
[7] S. L. Pan & J. N. Yang‚ Another simple and elementary proof of the classical isoperimetric inequality‚ Chinese Quarterly J. of Math.‚ 20:2(2005)‚ 128-130.
[8] 潘生亮 切线极坐标的一个应用 华东师范大学自然科学学报 2003 年第一期 13－ 16.
[9] S. L. Pan‚ On a new curve evolution problem in the plane‚ Topology and Geometry:
Commemorating SISTAG‚ A. J. Berrick‚ M. C. Leung & X. W. Xu‚ editors‚ Contemp. Math. 314‚ AMS‚ Providence‚ 2002‚ 209-217.
[11] S. L. Pan‚ Local solvability of a system of nonlinear differentia-integral equations‚ Chinese J. Contemporary Math. Vol. 22‚ No. 2 (2001)‚ 167-176.
[12] 潘生亮 一个非线性微分积分方程组的局部可解性 数学年刊 22A(2001)‚ 179-188.
(三)
時間：98 年 3 月 26 日至 4 月 2 日訪問上海期間，於 3 月 31 日上午訪問上海復旦大學數學系 忻元龍(Xin, Yuanlong)教授。之前就和忻元龍教授透過電子郵件討論，因為閱讀他的論文與書 籍，覺的可以和他討論，所以有如下的信件
It is great that you extend the Bernstein type theorems in many ways. I am happy to report you that I am working the Bernstein type theorems in another direction: from surfaces in R^3 with H=0 to H is L-2 integrable and finite total curvature. The only entire minimal graphs in R^3 are planes. From topological point of view, the entire graph can be treated as 1. simply connected, 2. one embedded end, 3. one embedded end with curvature condition. We have 3 Bernstein type theorems as attached files.
I am interesting your theorem 3 Theorem 3 Let M be a minimal n-submanifold in Rn+p with flat normal bundle and positive w-function. If M has finite total curvature, then M is totally
geodesic. in Bernstein type theorems with flat normal bundle Calc. Var. (2006) 26(1): 57–67.
Is it possible to relax H=0 to H is L-2 integrable and finite total curvature as I did before? Hope the information can entertain you. Your opinion is much welcomed.
因此該日我們討論了一個早上，結束時跟忻教授提及和洪家興是舊識，所以便一同共進 午餐，餐後又至復旦大學剛蓋好的大樓「光華樓」中的星光咖啡廳繼續討論。
忻元龍教授著作
Xin, Yuanlong Mean curvature flow with convex Gauss image. Chin. Ann. Math. Ser. B 29 (2008), no. 2, 121--134. (Reviewer: Valentino Tosatti) 53C44
Xin, Yuanlong Rigidity and mean curvature flow via harmonic Gauss maps. Front. Math. China 1 (2006), no. 3, 325--338. (Reviewer: Frédéric Robert) 53A10 (53C44 53C50)
Xin, Yuanlong Ricci curvature and fundamental group. Chinese Ann. Math. Ser. B 27 (2006), no. 2, 113--120. (Reviewer: María J. Druetta) 53C21 (20F34)
Xin, Yuanlong Minimal submanifolds and related topics. Nankai Tracts in Mathematics, 8. World Scientific Publishing Co., Inc., River Edge, NJ, 2003. viii+262 pp. ISBN: 981-238-687-4 (Reviewer: Andrew Bucki) 53C42 (53-02 53A10 53C40)
277--292. (Reviewer: Yu Xin Dong) 53C43 (32Q15 58E20)
Xin, Yuanlong Geometry of harmonic maps. Progress in Nonlinear Differential Equations and their Applications, 23. Birkhäuser Boston, Inc., Boston, MA, 1996. x+241 pp. ISBN: 0-8176-3820-2 (Reviewer: Martin A. Guest) 58E20
(四)
