平面上一种保长度曲线流
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摘要
本文主要研究平面上一种保持长度不变的曲线流,即令X(u,t):[a,b]×[0,∞)→R2是平面上一族闭曲线,X(u,0)=X0(u)是一条严格凸的平面闭曲线.考虑如下发展问题：
我们将证明在这种流下,曲线的周长不变,所围区域面积增大,曲线越来越圆.最后我们证明在“C∞”度量下,当t趋向于无穷大时,极限曲线是一个有限圆周(即具有有限半径的圆).更进一步地,如果初始曲线是常宽曲线,那么曲线在这种流下始终保持常宽,并且宽度不变.
This paper deals with a convex curve evolution problem in the plane. Let X(u, t):[a, b]×[0,∞)→R2 be a family of closed planer curves with X(u,0)= X0(u) being a closed, strictly convex curve. Consider the following problem:
We will prove this flow will preserve the perimeter of the evolving curve but enlarge the area it bounds and make the evolving curve more and more circular during the evolution process. And finally, as the time t goes to infinity, the limiting curve will be a finite circle (i.e., a circle with finite radius) in the C∞metric.Further more, the width of the curve keeps constant during the evolution if the initial curve has constant width, and the width is the same as the initial one.
我们将证明在这种流下,曲线的周长不变,所围区域面积增大,曲线越来越圆.最后我们证明在“C∞”度量下,当t趋向于无穷大时,极限曲线是一个有限圆周(即具有有限半径的圆).更进一步地,如果初始曲线是常宽曲线,那么曲线在这种流下始终保持常宽,并且宽度不变.
This paper deals with a convex curve evolution problem in the plane. Let X(u, t):[a, b]×[0,∞)→R2 be a family of closed planer curves with X(u,0)= X0(u) being a closed, strictly convex curve. Consider the following problem:
We will prove this flow will preserve the perimeter of the evolving curve but enlarge the area it bounds and make the evolving curve more and more circular during the evolution process. And finally, as the time t goes to infinity, the limiting curve will be a finite circle (i.e., a circle with finite radius) in the C∞metric.Further more, the width of the curve keeps constant during the evolution if the initial curve has constant width, and the width is the same as the initial one.
引文
[1]B. Andrews, Evolving convex curves, Calc. Var.& PDEs,7(1998),315-371.
[2]M. Athanassenas, Volume-preserving mean curvature flow of rotationally symmetric surfaces. Comment. Math. Helv.72,52-66(1997).
[3]K. S. Chou, X. P. Zhu, The Curve Shortening Problem, Chapman and Hall/CRC,2001.
[4]B. Chow, L. P. Liou, D. H. Tsai, Expanding of embedded curves with turning angle greater than-π, Invent. Math.123(1996),415-429.
[5]B. Chow, P. Lu, L. Ni, Hamilton's Ricci Flow, Beijing:Science Press; Provi-dence, R.I.:American Mathematical Society,2006.
[6]B. Chow, D. H. Tsai, Geometric expanding of convex plane curves, J. Diff. Geom.44(1996),312-330.
[7]K. Ecker, Regularity theory for mean curvature flow. Progress in Nonlinear Differential Equations and their Applications,57. Birkhauser Boston, Inc., Boston, MA,2004.
[8]M. E. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J.50(1983),1225-1229.
[9]M. E. Gage, Curve shortening makes convex curves circular, Invent. Math. 76(1984),357-364.
[10]M. E. Gage, On an area-preserving evolution equation for plane curves, in "Nonlinear Problems in Geometry" (D. M. DeTurck edited), Contemp. Math. Vol.51(1986),51-62.
[11]M. E. Gage, Curve shortening on surfaces, Ann. Scient. Ec. Norm. Sup. 23(1990),229-256.
[12]M. E. Gage, R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom.23(1986),69-96.
[13]M. Grayson,The heat equation shrinks embedded plane curve to round points, J. Diff. Geom.26(1987),285-314.
[14]G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom.20(1984),237-266.
[15]G. Huisken, The volume preserving mean curvarure flow. J. Reine Angew. Math.382,35-48(1987).
[16]J. A. McCoy, The surface area preserving mean curvature flow. Asian J. Math. 7(1),7-30(2003).
[17]J.A.McCoy, Mixed volume preserving curvature flows. Calc. Var. PDEs 24(2), 131-154(2005).
[18]R. Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly,86(1979),1-29.
[19]S. L. Pan, A note on the general curve flows, J. of Math. Study,33(2000), 17-26.
[20]S. L. Pan, J. N. Yang. On a non-local perimeter-preserving curve evolution problem for convex plane curves, Manuscripta Math.127(2008),469-484.
[21]S. L. Pan, H. Zhang, A reverse isoperimetric inequality for convex plane curves, Bertrage zur Algebra und Geometrie,48(2007),303-308.
[22]R. Schneider, Convex Bodies:the Brunn-Minkowski Theory, Cambridge Uni-versity Press, Cambridge-New York,1993.
[23]D. H. Tsai, Blowup and convergence of expanding immersed convex plane curves, Comm. Anal. Geom.8(2000),761-794.
[24]D. H. Tsai, On the formation of singularities in the curve expanding flow, Calc. Var. & PDEs,14(2002),385-398.
[25]D. H. Tsai, Asymptotic closeness to limiting shapes for expanding embedded plane curves, Invent. Math.162(2005),473-492.
[26]K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math.38(1985),867-882.
[27]J. Urbas, Convex curves moving homothetically by negative powers of their curvature, Asian J. Math.3(1999),635-658.
[28]M. T. Wang, Lectures on mean curvature flows in higher codimensions, in Handbook of Geometric Analysis, Vol. Ⅰ (edited by L. Z. Ji, P. Li, R. Schoen & L. Simon), pp.525-543, Higher Education Press,2008.
[29]B. White, Evolution of curves and surfaces by mean curvature, Proc. of the ICM 2002, Vol.I,525-538.
[30]X. P. Zhu, Lectures on mean cuvature flows. AMS/IP Studies in Advanced Mathematics, vol.32. American Mathematical Society/International Press, Providence/Somerville(2002)
[2]M. Athanassenas, Volume-preserving mean curvature flow of rotationally symmetric surfaces. Comment. Math. Helv.72,52-66(1997).
[3]K. S. Chou, X. P. Zhu, The Curve Shortening Problem, Chapman and Hall/CRC,2001.
[4]B. Chow, L. P. Liou, D. H. Tsai, Expanding of embedded curves with turning angle greater than-π, Invent. Math.123(1996),415-429.
[5]B. Chow, P. Lu, L. Ni, Hamilton's Ricci Flow, Beijing:Science Press; Provi-dence, R.I.:American Mathematical Society,2006.
[6]B. Chow, D. H. Tsai, Geometric expanding of convex plane curves, J. Diff. Geom.44(1996),312-330.
[7]K. Ecker, Regularity theory for mean curvature flow. Progress in Nonlinear Differential Equations and their Applications,57. Birkhauser Boston, Inc., Boston, MA,2004.
[8]M. E. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J.50(1983),1225-1229.
[9]M. E. Gage, Curve shortening makes convex curves circular, Invent. Math. 76(1984),357-364.
[10]M. E. Gage, On an area-preserving evolution equation for plane curves, in "Nonlinear Problems in Geometry" (D. M. DeTurck edited), Contemp. Math. Vol.51(1986),51-62.
[11]M. E. Gage, Curve shortening on surfaces, Ann. Scient. Ec. Norm. Sup. 23(1990),229-256.
[12]M. E. Gage, R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom.23(1986),69-96.
[13]M. Grayson,The heat equation shrinks embedded plane curve to round points, J. Diff. Geom.26(1987),285-314.
[14]G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom.20(1984),237-266.
[15]G. Huisken, The volume preserving mean curvarure flow. J. Reine Angew. Math.382,35-48(1987).
[16]J. A. McCoy, The surface area preserving mean curvature flow. Asian J. Math. 7(1),7-30(2003).
[17]J.A.McCoy, Mixed volume preserving curvature flows. Calc. Var. PDEs 24(2), 131-154(2005).
[18]R. Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly,86(1979),1-29.
[19]S. L. Pan, A note on the general curve flows, J. of Math. Study,33(2000), 17-26.
[20]S. L. Pan, J. N. Yang. On a non-local perimeter-preserving curve evolution problem for convex plane curves, Manuscripta Math.127(2008),469-484.
[21]S. L. Pan, H. Zhang, A reverse isoperimetric inequality for convex plane curves, Bertrage zur Algebra und Geometrie,48(2007),303-308.
[22]R. Schneider, Convex Bodies:the Brunn-Minkowski Theory, Cambridge Uni-versity Press, Cambridge-New York,1993.
[23]D. H. Tsai, Blowup and convergence of expanding immersed convex plane curves, Comm. Anal. Geom.8(2000),761-794.
[24]D. H. Tsai, On the formation of singularities in the curve expanding flow, Calc. Var. & PDEs,14(2002),385-398.
[25]D. H. Tsai, Asymptotic closeness to limiting shapes for expanding embedded plane curves, Invent. Math.162(2005),473-492.
[26]K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math.38(1985),867-882.
[27]J. Urbas, Convex curves moving homothetically by negative powers of their curvature, Asian J. Math.3(1999),635-658.
[28]M. T. Wang, Lectures on mean curvature flows in higher codimensions, in Handbook of Geometric Analysis, Vol. Ⅰ (edited by L. Z. Ji, P. Li, R. Schoen & L. Simon), pp.525-543, Higher Education Press,2008.
[29]B. White, Evolution of curves and surfaces by mean curvature, Proc. of the ICM 2002, Vol.I,525-538.
[30]X. P. Zhu, Lectures on mean cuvature flows. AMS/IP Studies in Advanced Mathematics, vol.32. American Mathematical Society/International Press, Providence/Somerville(2002)