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There is a description a plane curve called its Cesàro equation, in which the curvature is given as a function $k$ of arc length $s$. Suppose that $k$'s domain is $(0,1]$. Then it should be clear that, given one point and a tangent vector at that point, the corresponding curve $\mathbf{r}(s)$ exists and is unique for $s\in[0,1]$. (Even if the function $k$ misbehaves at $s=0$, the limit $\mathbf{r}(0)$ exists.)

Define a the curve C whose Cesàro equation for $s\in(0,1]$ is $k=1/s^2$, at any desired orientation with (arbitrarily, but for concreteness) $\mathbf{r}(0)=(1,0)$, and position. Construct a line L whose distance from its tangent vector upward at $\mathbf{r}(0)$ is 1. s=0$. Form a surface of revolution S by revolving C about Lthe $y$ axis. S is compact, and the locus $s=0$ is a unit circle forming a boundary of S.

The Gaussian curvature $\kappa$ is equal in magnitude to the product of the curvatures along the two principal axes. The curvature along the azimuthal axis is 1 for $s=0$, so the Gaussian curvature behaves asymptotically like $-1/s^2$ |\kappa|\sim 1/s^2$ as $s\rightarrow 0$.
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There is a description a plane curve called its Cesàro equation, in which the curvature is given as a function $k$ of arc length $s$. Suppose that $k$'s domain is $(0,1]$. Then it should be clear that, given one point and a tangent vector at that point, the corresponding curve $\mathbf{r}(s)$ exists and is unique for $s\in[0,1]$. (Even if the function $k$ blows up misbehaves at $s=0$, the limit $\mathbf{r}(0)$ exists.)

Define a curve C whose Cesàro equation for $s\in(0,1]$ is $k=1/s^2$, at any desired orientation and position. Construct a line L whose distance from $\mathbf{r}(0)$ is 1. Form a surface of revolution S by revolving C about L. S is compact, and the locus $s=0$ is a boundary of S.

The Gaussian curvature is equal in magnitude to the product of the curvatures along the two principal axes. The curvature along the azimuthal axis is 1 for $s=0$, so the Gaussian curvature behaves asymptotically like $-1/s^2$ as $s\rightarrow 0$.
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There is a description a plane curve called its Cesàro equation, in which the curvature is given as a function $k$ of arc length $s$. Suppose that $k$'s domain is $(0,1]$. Then it should be clear that, given one point and a tangent vector at that point, the corresponding curve $\mathbf{r}(s)$ exists and is unique for $s\in[0,1]$. (Even if the function $k$ blows up at $s=0$, the limit $\mathbf{r}(0)$ exists.)

Define a curve C whose Cesàro equation for $s\in(0,1]$ is $k=1/s^2$, at any desired orientation and position. Construct a line L whose distance from $\mathbf{r}(0)$ is 1. Form a surface of revolution S by revolving C about L. S is compact, and the locus $s=0$ is a boundary of S.

The Gaussian curvature is equal in magnitude to the product of the curvatures along the two principal axes. The curvature along the azimuthal axis is 1 for $s=0$, so the Gaussian curvature behaves asymptotically like $-1/s^2$ as $s\rightarrow 0$.