“I know a place, where the grass is really greener,” sings pop star Katy Perry in her 2010 hit “California Gurls.”

On a November morning in Perry’s hometown of Santa Barbara, Calif., that seemed to be the case. On a gated estate spanning more than 2 acres in the coveted enclave of Montecito, the only sound was the gurgling of the fountains. Lime trees and rosebushes ringed a large, grassy lawn. The spacious, circa-1930 house was backed by green mountains, and the Pacific Ocean and Channel Islands could be seen through a gap in the towering hedges encircling the compound.