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July  2014, 13(4): 1641-1652. doi: 10.3934/cpaa.2014.13.1641

The classification of constant weighted curvature curves in the plane with a log-linear density

Doan The Hieu 1, and  Tran Le Nam 2,

1. 

Departement of Mathematics, College of Education, Hue University, Hue, Vietnam
2. 

Departement of Mathematics,Dong Thap University, Dong Thap, Vietnam

Received  May 2013 Revised  January 2014 Published  February 2014

In this paper, we classify the class of constant weighted curvature curves in the plane with a log-linear density, or in other words, classify all traveling curved fronts with a constant forcing term in $R^2.$ The classification gives some interesting phenomena and consequences including: the family of curves converge to a round point when the weighted curvature of curves (or equivalently the forcing term of traveling curved fronts) goes to infinity, a simple proof for a main result in [13] as well as some well-known facts concerning to the isoperimetric problem in the plane with density $e^y.$
Keywords: planes with a log-linear density, self similar translator, traveling curved front, weighted curvature., Curve flow with a forcing term.
Mathematics Subject Classification: Primary 53C44, 53A04; Secondary 53C21, 35J6.
Citation: Doan The Hieu, Tran Le Nam. The classification of constant weighted curvature curves in the plane with a log-linear density. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1641-1652. doi: 10.3934/cpaa.2014.13.1641
References:
[1]

M. A. S. Aarons, Mean curvature flow with a forcing term in Minkowski space, Calc. Var. Partial Differential Equations, 25 (2006), 205-246. doi: 10.1007/s00526-005-0351-8.  Google Scholar

[2]

C. Carroll, A. Jacob, C. Quinn and R. Walters, The isoperimetric problem on planes with density, Bull. Aust. Math. Soc., 78 (2008), 177-197. doi: 10.1017/S000497270800052X.  Google Scholar

[3]

A. Cañete, M. Miranda and D. Vittone, Some isoperimetric problems in planes with density, J. Geo. Anal., 20 (2010), 243-290. doi: 10.1007/s12220-009-9109-4.  Google Scholar

[4]

I. Corwin, N. Hoffman, S. Hurder, V. Sesum and Y. Xu, Differential geometry of manifolds with density, Rose-Hulman Und. Math. J., 7 (2006). Google Scholar

[5]

I. Corwin and F. Morgan, The Gauss-Bonnet formula on surfaces with densities, Involve, 4 (2011), 199-202. doi: 10.2140/involve.2011.4.199.  Google Scholar

[6]

J. Dahlberg, A. Dubbs, E. Newkirk and H. Tran, Isoperimetric regions in the plane with density $r^p$, New York J. Math., 16 (2010) 31-51.  Google Scholar

[7]

K. Ecker and G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys., 135 (1991), 595-613.  Google Scholar

[8]

D. T. Hieu and N. M. Hoang, Ruled minimal surfaces in $R^3$ with density $e^z$, Pacific J. Math., 243 (2009), 277-285. doi: 10.2140/pjm.2009.243.277.  Google Scholar

[9]

D. T. Hieu, Some calibrated surfaces in manifolds with density, J. Geom. Phys., 61 (2011), 1625-1629. doi: 10.1016/j.geomphys.2011.04.005.  Google Scholar

[10]

G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. PDE, 8 (1999), 1-14. doi: 10.1007/s005260050113.  Google Scholar

[11]

G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70. doi: 10.1007/BF02392946.  Google Scholar

[12]

H. Jian, H. Ju, Y. Liu and W. Sun, Symmetry of translating solutions to mean curvature flows, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 2006-2016. doi: 10.1016/S0252-9602(10)60191-9.  Google Scholar

[13]

H. Jian, H. Ju, Y. Liu and W. Sun, Traveling fronts of curve flow with external force field, Commun. Pure Appl. Anal., 9 (2010), 975-986. doi: 10.3934/cpaa.2010.9.975.  Google Scholar

[14]

H. Ju, J. Lu and H. Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space, Commun. Pure Appl. Anal., 9 (2010), 963-973. doi: 10.3934/cpaa.2010.9.963.  Google Scholar

[15]

Q. Maurmann and F. Morgan, Isoperimetric comparison theorems for manifolds with density, Calc. Var. PDE, 36 (2009), 1-5. doi: 10.1007/s00526-008-0219-9.  Google Scholar

[16]

F. Morgan, Manifolds with density, Notices Amer. Math. Soc., 52 (2005), 853-858.  Google Scholar

[17]

F. Morgan, Myers' Theorem with density, Kodai Math. J., 29 (2006), 454-460. doi: 10.2996/kmj/1162478772.  Google Scholar

[18]

F. Morgan, Geometric Measure Theory: a Beginner's Guide, $4^{th}$ edition, Academic Press, 2009.  Google Scholar

[19]

F. Morgan, Manifolds with density and Perelman's proof of the Poincaré Conjecture, Amer. Math. Monthly, 116 (2009), 134-142. doi: 10.4169/193009709X469896.  Google Scholar

[20]

H. Ninomiya and M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force, Free boundary problems: theory and applications, I (Chiba, 1999). 206-221, GAKUTO Internat. Ser. Math. Sci. Appl., 13, Gakkotosho, Tokyo, 2000.  Google Scholar

[21]

C. Rosales, A. Cañete, V. Bayle and F. Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. PDE, 31 (2008), 27-46. doi: 10.1007/s00526-007-0104-y.  Google Scholar

show all references

References:
[1]

M. A. S. Aarons, Mean curvature flow with a forcing term in Minkowski space, Calc. Var. Partial Differential Equations, 25 (2006), 205-246. doi: 10.1007/s00526-005-0351-8.  Google Scholar

[2]

C. Carroll, A. Jacob, C. Quinn and R. Walters, The isoperimetric problem on planes with density, Bull. Aust. Math. Soc., 78 (2008), 177-197. doi: 10.1017/S000497270800052X.  Google Scholar

[3]

A. Cañete, M. Miranda and D. Vittone, Some isoperimetric problems in planes with density, J. Geo. Anal., 20 (2010), 243-290. doi: 10.1007/s12220-009-9109-4.  Google Scholar

[4]

I. Corwin, N. Hoffman, S. Hurder, V. Sesum and Y. Xu, Differential geometry of manifolds with density, Rose-Hulman Und. Math. J., 7 (2006). Google Scholar

[5]

I. Corwin and F. Morgan, The Gauss-Bonnet formula on surfaces with densities, Involve, 4 (2011), 199-202. doi: 10.2140/involve.2011.4.199.  Google Scholar

[6]

J. Dahlberg, A. Dubbs, E. Newkirk and H. Tran, Isoperimetric regions in the plane with density $r^p$, New York J. Math., 16 (2010) 31-51.  Google Scholar

[7]

K. Ecker and G. Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys., 135 (1991), 595-613.  Google Scholar

[8]

D. T. Hieu and N. M. Hoang, Ruled minimal surfaces in $R^3$ with density $e^z$, Pacific J. Math., 243 (2009), 277-285. doi: 10.2140/pjm.2009.243.277.  Google Scholar

[9]

D. T. Hieu, Some calibrated surfaces in manifolds with density, J. Geom. Phys., 61 (2011), 1625-1629. doi: 10.1016/j.geomphys.2011.04.005.  Google Scholar

[10]

G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. PDE, 8 (1999), 1-14. doi: 10.1007/s005260050113.  Google Scholar

[11]

G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70. doi: 10.1007/BF02392946.  Google Scholar

[12]

H. Jian, H. Ju, Y. Liu and W. Sun, Symmetry of translating solutions to mean curvature flows, Acta Math. Sci. Ser. B Engl. Ed., 30 (2010), 2006-2016. doi: 10.1016/S0252-9602(10)60191-9.  Google Scholar

[13]

H. Jian, H. Ju, Y. Liu and W. Sun, Traveling fronts of curve flow with external force field, Commun. Pure Appl. Anal., 9 (2010), 975-986. doi: 10.3934/cpaa.2010.9.975.  Google Scholar

[14]

H. Ju, J. Lu and H. Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space, Commun. Pure Appl. Anal., 9 (2010), 963-973. doi: 10.3934/cpaa.2010.9.963.  Google Scholar

[15]

Q. Maurmann and F. Morgan, Isoperimetric comparison theorems for manifolds with density, Calc. Var. PDE, 36 (2009), 1-5. doi: 10.1007/s00526-008-0219-9.  Google Scholar

[16]

F. Morgan, Manifolds with density, Notices Amer. Math. Soc., 52 (2005), 853-858.  Google Scholar

[17]

F. Morgan, Myers' Theorem with density, Kodai Math. J., 29 (2006), 454-460. doi: 10.2996/kmj/1162478772.  Google Scholar

[18]

F. Morgan, Geometric Measure Theory: a Beginner's Guide, $4^{th}$ edition, Academic Press, 2009.  Google Scholar

[19]

F. Morgan, Manifolds with density and Perelman's proof of the Poincaré Conjecture, Amer. Math. Monthly, 116 (2009), 134-142. doi: 10.4169/193009709X469896.  Google Scholar

[20]

H. Ninomiya and M. Taniguchi, Traveling curved fronts of a mean curvature flow with constant driving force, Free boundary problems: theory and applications, I (Chiba, 1999). 206-221, GAKUTO Internat. Ser. Math. Sci. Appl., 13, Gakkotosho, Tokyo, 2000.  Google Scholar

[21]

C. Rosales, A. Cañete, V. Bayle and F. Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. PDE, 31 (2008), 27-46. doi: 10.1007/s00526-007-0104-y.  Google Scholar

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