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The classification of constant weighted curvature curves in the plane with a log-linear density
1. | Departement of Mathematics, College of Education, Hue University, Hue, Vietnam |
2. | Departement of Mathematics,Dong Thap University, Dong Thap, Vietnam |
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. |
[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. |
[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. |
[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). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
F. Morgan, Manifolds with density, Notices Amer. Math. Soc., 52 (2005), 853-858. |
[17] |
F. Morgan, Myers' Theorem with density, Kodai Math. J., 29 (2006), 454-460.
doi: 10.2996/kmj/1162478772. |
[18] |
F. Morgan, Geometric Measure Theory: a Beginner's Guide, $4^{th}$ edition, Academic Press, 2009. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
F. Morgan, Manifolds with density, Notices Amer. Math. Soc., 52 (2005), 853-858. |
[17] |
F. Morgan, Myers' Theorem with density, Kodai Math. J., 29 (2006), 454-460.
doi: 10.2996/kmj/1162478772. |
[18] |
F. Morgan, Geometric Measure Theory: a Beginner's Guide, $4^{th}$ edition, Academic Press, 2009. |
[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. |
[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. |
[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. |
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