I think there is such a minimal surface with area less than pi(1-r^2). Look at the catenoid formed by rotation of y a(cosh a) about the x axis where a is small the the distance from the origin will be 2a or more with the minimum at x=0. Now it will intersect the circle at a value of x less than (ln (a^-1))^2 actually well before that but clearly e^((log a)^2) is a^(ln a) (note this is a small number to a large negative power which gives a large number which is bigger than a 2/a which is all I need (note I am also overestimating the length by assuming the radius where the catenoid intersects the sphere is one)I need one) I estimate the area with two cones using the formula 2(pi)rs which will have a greater area than the minimal surface and get an upper bound of 2(ln a)^2 - a small factor due to the circle at the origin which I ignore since the number is small enough already not to mention I am already grossly overestimating the length. So I get an upper bound of the area of 2(ln a)^2 which is less than pi(1-4a^2).
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I think there is such a minimal surface with area less than pi(1-r^2). Look at the catenoid formed by rotation of y a(cosh a) about the x axis where a is small the the distance from the origin will be 2a or more with the minimum at x=0. Now it will intersect the circle at a value of x less than (ln (a^-1))^2 actually well before that but clearly e^((log a)^2) is a^(ln a) which is bigger than a which is all (note I am also overestimating the length by assuming the radius where the catenoid intersects the sphere is one)I need I estimate the area with two cones using the formula 2(pi)rs which will have a greater area than the minimal surface and get an upper bound of 2(ln a)^2 - a small factor due to the circle at the origin which I ignore since the number is small enough already not to mention I am already overestimating the length. So I get an upper bound of the area of 2(ln a)^2 which is less than pi(1-4a^2). |
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