# AdS soliton The AdS soliton, found by [[1998#Horowitz, Myers]], is a solution to vacuum Einstein's equation with a negative cosmological constant, i.e., it is asymptotically AdS. The conformal boundary is the $d$-dimensional Minkowski spacetime $\mathbb{R}^{d-1,1}$. Each of these $d$ flat dimensions can be compactified to become a circle (including the time-direction after we go to Euclidean signature). Their metric is given by$d s^2=\frac{r^2}{\ell^2}\left[-d t^2+\left(1-\frac{r_0^{p+1}}{r^{p+1}}\right) d \tau^2+\left(d x^i\right)^2\right]+\left(1-\frac{r_0^{p+1}}{r^{p+1}}\right)^{-1} \frac{\ell^2}{r^2} d r^2,$where $i\in\{1,\dots,p-1\}$ and $l$ is the AdS length. We can see that the spatial circle along $\tau$ shrinks to zero at $r=r_0$. To ensure regularity of the geometry at $r_0$, we also need to choose the periodicity of $\tau$ to be $\beta=4 \pi l^2 /(p+1) r_0$. The energy, computed using the method of [[1995#Hawking, Horowitz|Hawking and Horowitz]] and with the reference metric chosen to be Poincare AdS, is given by$E=-\frac{V_{p-1} \ell^p}{16 \pi G_{p+2} \beta^p}\left(\frac{4 \pi}{p+1}\right)^{p+1},$where $V_{p-1}$ is the volume of $x^i$ directions. It is a very interesting solution because it has lower energy than pure AdS, which is maximally symmetric and thus expected to be the ground state. With some numerical evidence, this led to the [[0407 Horowitz-Myers conjecture|HM conjecture]], stating that the AdS solition has the lowest energy among all solutions satisfying the same boundary conditions.